Tjyylli

Society as we know it certainly could NOT function if this were the case. A couple of issues that immediately occurred to me, even more fundamental than retention of property (and retention of property is huge!):

Children would be extremely dangerous to be around!

Adults are significantly stronger than children. Imagine if a mother and her one-year-old son happened to swap bodies. Suddenly you might have a one-year-old's personality and (lack of) wisdom in the body of a grown woman, and the mother in a helpless and weak body. The woman-bodied child might accidentally crush the child-bodied mother, or fall into a fire, etc. The mother would be physically unable to prevent these problems. Or imagine an angry four-year-old, still smarting from being told it was naughty to hit people, and suddenly in his father's body and capable of actually hurting people!

Power could be abused forever

Assuming the human race could survive at all... If proximity and simultaneous sleep could potentially be used to swap bodies, relative immortality would be possible for some. Enclaves could be established where only those of a moderate age and reasonable health were allowed to sleep, so the only body you would "risk" waking up in would be of acceptable quality. As someone "aged out", they could kidnap a young body, and lock the young body and the aging individual in an even more isolated cell for sleep-periods. When the locked-up kidnapped body can deliver the right password-type proofs that it's got the right person in it, you know the transfer is successful, and the old body may be eliminated.

Nothing like the world we currently live in would be possible.

I actually think large cities could function but they would be very different to ours.

Your house would be near a cluster of sleeping rooms and would have a password entry since keys are not going to work. Clusters of sleeping rooms are at least 5 miles apart and probably need some excellent public transit options. That is probably the hardest part of making a city and is probably going to have assigned sleep times to help allow reuse. People would be assigned to these facilities based on age, gender, physical condition and so forth so you keep a similar body at least.

There would be numerous government centers that would give you a daily ID after you give a password so you can have an identity and legal agreements based on that. You have to show it to get into your job and you have to put its info into your phone to receive calls.

There will be huge penalties for sleeping in random places or at random times, as that threatens the system.

Children will be communal. Since no one really has DNA that is theirs and letting children swap with adults would be a disaster children will be raised in group facilities that are at least 5 miles away from any other places people might sleep. Being a parent will be a job like any other since little kids keep swapping around getting attached to any single one is harder.

Prison could simply be being stuck in disabled bodies, no matter how evil you are if you are paralyzed you aren't going to do much damage.

Coffee and other stimulants are practically mandatory if you are feeling tired while out and about so they will possibly be government dispensed. Health care is also going to be communal since one person's illness is now literally everyone's problem.

Immortality is possible and that changes things a lot. An old person just needs to set it up so they sleep with young people and boom they are young again. This could be the worst crime possible or a totally legal thing that's expensive but either way it would allow all kinds of radical changes to the world. Either way these cities could possibly have the same people in charge forever and that could help with stabilizing things against the chaos this causes.

Class Inequality
You would likely end up with sectarian societies, one of which would be elites who create a "gated community" city that would exploit the labor of the lower class, but who would prevent those classes from accessing their community.

Religious Implications
If their society is anything like ours, there would likely be monastical living situations based on a religious compulsion. People who shared these values would likely live as monks and nuns, since sharing bodies between sexes would be considered improper. Also, if there was something resembling a patriarchal power imbalance, men or women might not want the other to share in their experience.

Values
This society, like ours, would be drawn towards empathy if not for the constant insistence on segregation of experience. Cultures that are empathetic would be ones that perhaps are not as technologically advanced, and therefore unable to travel the 5 miles to and from resources provided by others. The value of the individual would grow as technology advanced far enough to allow it to do so, and the value of community would be inversely proportionate to this.

Loophole
You could consider creating the condition that the "size" of a person's mind (their experiences, intelligence, etc.) only transfers to a suitable host with a brain that allows for the same level of functionality. This would solve the problem of an isolated family. The mother and father might switch minds regularly, but the children might not until they had reached a certain age, and upon approaching adulthood, could possibly then switch with the parents. It would at least prevent a fetus from switching minds with its mother, which makes sense since a fetus' brain is not physically developed to the point of sustaining an adult's mind. This condition would also create some interesting characters--geniuses or those with broader experiences might be quite lonely waking up in the same body. The consistency of their physical appearance might make them natural leaders. In this scenario there would most likely be a group of leaders who wake up in different bodies but are consistently the same 10-20 people, revered for their experience and intellect. A twist to this story might happen when someone who doesn't consider themselves very intelligent suddenly finds themselves awake in the bodies of one of these elders.

It would be very difficult for society to function correctly.

Imagine jumping into the body of a diabetic,someone with peanut allergies or any number of chronic medical conditions that an be managed by those that know they have them and what to do about it.

Randomly awakening in a body and not knowing that before breakfast you need to medicate or your new body will suffer serious consequences. Even something simple like eyeglasses without them a person may be unable to see properly, knowing where you left your pair the night before is important if your prescription is high enough.

Jack Chalker seems to think that it might be possible - he postulates a similar society in the second book of his Four Lords of the Diamond series, Cerberus: A Wolf in the Fold.

His society is somewhat more controlled than most modern societies, and has a method of identifying a person mentally - that is, the mind that's in the body. In general, there are two classes of person - those that keep the same job regardless of the body they're in (i.e., the job stays with the mind), and those that keep the same job regardless of the mind that's in the body (i.e., the job stays with the body). Most white-collar jobs are in the first class; undesirable jobs, and (in his society) host-motherhood, are in the second, and it's rare for people in the first class to associate with people in the second (because what happens if a first-class swaps with a second-class?).

Current society? No, but I think that this type of system would actually promote people to live very similar, structured communal lives.

Imagine you wake up in someone's body, but that person didn't have any food in "his" house the day before. Now you go hungry, right? Suddenly it also dawns on you, how do you pay? Any physical paying methods would be on your previous body, and every time you go to sleep that would change again.

This means that every person would make sure the basics of life would always be available nearby, if only because if they don't do it someone else might also not do it. This would also mean that going drunk before going to sleep would be frowned upon, and people might actually be arrested and chemically kept awake until they aren't drunk or whatever anymore for the sake of whichever poor soul has to inhabit that person.

Communities would still be able to have their own houses and things, but the sleeping arrangements would all be located in one communal area. This way you know where you'll wake up, and after waking up you can move to your own house with your own things, live your life normally and then go back to the communal area's for sleeping. If this is how it's always been then monetary things might never really be invented. If it suddenly happens to our current world then we would try to use electronics to keep track of our money and to secure our homes. A memorized code can be taken with you when you go to sleep, the keys to your house and car can't.

Younger people would prefer not to wake up in an old body that is about to die or might never wake up. It is likely that bodies in general are going to get tatoo's to indicate when they were born, and thrown into one large group of similar age. These groups will always go to sleep in one area, away from other age groups. This way the group as a whole will grow old and die. There will definitely be people who try to abuse it and go to sleep nearby a younger generation group. This can be countered through a set of standard questions that each group gets upon getting up (easier when computer technology comes around), to see if everyone is accounted for. If someone from an older age group has managed to get inside the body of a younger age group he won't know the correct answers to identify himself as a specific person from that group and a search will happen for the person who does have the correct answer, after which they are forced to swap back again.

This communal thinking will have to stretch to everything, including children. If you get knocked up one day, you'll have a good chance you'll never even see your own pregnant body again! Children would have to be raised in communities as well, as there's no real telling who is the mother or father. The person who finds out if you got knocked up is going to be inhabiting it months down the line!

Assuming the switch only occurs while you are asleep, and assuming you want people to have private property it could go something like this.

Me and my neighbors all have a secure sleeping pod on the side of our
house. I go to sleep in it every night because I know that when I
wake up I will be in another pod and someone else will be in mine. I
made sure to put all my valuables in my house before I went to
bed. From the pod, the stranger will wake up, but they can't get into
my house or access my stuff because the pod is not in my house.
Usually they politely exit the pod and take a ride share back to their
house (ist just a few miles).

Every new body has its quirks, but I have been in thousands of bodies
before so I learn pretty quickly now. This one is very similar to one I was in last
month.

The touch screen in the pod I woke up in has the ride share app on it. I also call
for a ride. I pay the driver by telling him my account number. When
I get there my wife is waiting for me in a new body. I
don't really know what she originally looked like, but neither does she. I
greet her and verify her identity by using my co-name (a secret name
that is used only with another person or group)
she verifies her identity by responding with her correct co-name.

I have different co-names I use with different people depending on how much I
trust them.

I use my pin code to open my front door and get ready for work.

The kids are in a public boarding school. The school is outside the
city so they don't switch with any adults. The teachers sleeping area
is a few miles from the school. There are night time care-takers at
the school to help children. The caretakers never sleep at the school.
We Skype the kids during the week and visit
them during the day on weekends.

We don't know who our kids original parents were. They are not biologically
ours. And their minds are not from children who were biologically ours.
But we adopted them when they became old enough to be able to understand their
co-names. We had another child for a few months, but I think he was a bit too
young when we adopted him. One day he stopped identifying himself to us and
we never figured out how to find them.

I think this arrangement could allow a society to get reasonably big. But cities might be a little more spread out since you need separate areas for the children.

I think that one major change would be that the use of memorized access codes or account info would be more common since physical keys or cards wouldn't be useable after initially waking up. Also biometric identification would be totally useless.