While thinking about "how would it fit into modern society" is fun, what if we'll think about development of such species from ancient times?
Ancient times (humanity begins)
So here we are, small (let's say 4-15 person) groups who recivied souls (somehow). Which problems such group will likely face?
- Bodies may vary.
Yesterday you were able to go hunting, today you are much better at looking for babies and harveting berries. I have no idea about their psycology at that times, but idea of being genderless is likely: whatever, it is just process of creating children, no hard feelings. Monogamy is very unlikely too: it is not profitable for surviving anymore (if sex is still about biology and not souls, otherwise entire humanity will work different way starting from this point).
This also leads to learning of everything known regardless of anything: person should be able to use any body at its max abilities.
- Children problem.
As other answers have pointed out, children are dangerous now. Let somebody wake up in their body and they are useless for a day. Let very young baby wake up in adult body — they are useless again but also posess a great danger to others.
This could be solved by never sleeping at same time children do. Hopefully, your day still have around 24 hours and your fellow humans need about 8 hours of sleep everyday. In this case we can divide day into two part: 15-hours one for adults' sleep (exact time may vary, also different people may enter and leave sleep during it) and 8-9 hours when all adults don't sleep and children do. We also now have somebody awaken to keep an eye on your children to don't let them fall asleep.
Moving into adult-time sleep may occur in specific age or when one is skilled enough to do something useful. This include some interesting rites tho…
- Elders and deathly ill ones problem.
Things turn out sad this time. When some body becames too weak it will be better for spiece to get rid of it by exiling or even directly killing them. But who will take it to die within? It could be one of that: random unlucky person, somebody least useful (good for spiece 2x) or some children.
Why children you say? It will allow to maintain knowledge easily, without need to write down or even talk… and never develop speech nor modern society. We just hit dead end. Such society will exsist and will likely be found by researches in distant future ("Scientists have discovered group of persons whose souls must have seen 10000 years BC" - BBC from your world).
So some groups decided to put only full aged ones in elders bodies and force them to go away to die. Reason doesn't matter — some randomy appeared rite would be enough.
Also, rite to keep your todays' body in great form will be required because group will die really quickly.
Let's say at some point our last group develop speech and have to face an important problem: technologies became too complicated to be perfected by everyone and profession may require different body variations. What to do now?
Easy and efficient solution would be to let different masters live in separated villages/towns where exchange of souls is impossible. In order to exchange stuff they will have to meet in some kind of special place like giant market where all kind of buisness do occur: it also may include some common education schools. Some persons who wish to change their lives may also sleep there to exchange bodies with others from different towns. However, it will likely be punished by others due to requiment to teach them again, if they aren't children (which now have a way to get body sutable for profession of their dream!).
This model also may lead to some fun evolution where different towns develop their own different DNA samples (each town is a distinct population actually) resulting in specialized farmers, blacksmithes, miners and others, probably even ending in ones who differ a lot if this epoch lasts long enough.
As for government, it is not very likely to have one (and much harder to have more than one). Wars would became something insane due to constant exchange of bodies.
Such society will require strond punishment for betraying/cheating within one town, or it will fall apart. This require absolute trust at first, which potentially could be maintained on long term too. Religion would help a lot.
And to our older problems:
- Bodies may vary.
Nothing pretty much changes. Everybody in town is married by everybody, everybody master the same thing.
- Children problem.
And we still have same solution. Other one is impossible until we have a way of isolating children from overwatchers (while keeping them alive).
- Elders and deathly ill ones problem.
And here we have some positive changes: elders from different towns may live in separated one, helping each other. While it is still doesn't seem so great, it is at least sounds a bit more accepable, huh?
You have a great idea which nobody seems to implement yet? Well, we are in trouble here…
This society is very stable and can change hardly: pesonality is not valuable here, congrats on building communism. It may last many centuries or even few thousands of years until somebody comes to change it…
It is impossible to say how exactly will it begin. Probably, someday some crazy person finds enough followers on giant marketplace who agree to join him on everyday base is some crazy project like steam engine or something else which will knock the world over. I'm not going to describe it completely here (that would be interesting story to read for me tho), let's say that humanity begins to belive into innovations, and this means a lot. Some people begin to leave their towns to join new village of researchers… which probaly became rich pretty soon!
There are some critical things which could change your society a lot:
- Automated machines which can take care of children. They would allow to finally escape childrens problems and use humans much more efficiently.
- Shuffle inhibitors. If somebody manages to invent them and produce in huge portions, they will be able to insantly overcome a lot of complications, making it possible to build stable group which could became an army capable of acting hightly organized and capturing cities with ease. This may lead to a very long way to something like our society either in case of war or making them publicly available.
If last one don't happen, it is possible to survive within realtively small towns and without government: things like private property became impossible because you can't remember anything not guessable within reasonable time and there is no way to do 2FA. Of course, until you have
- USID — unique sould ID.
If there is a way to confirm that some body have somebody's soul, it becames possible to build all kind of advanced things we have. It will can not lead to any kind of war, making it possible to migrate into large low-trust cities where it is possible to mantain all kinds of property and access restrictions.
People will likely have something like "house", place where they can enjoy their favorite private things like videogames, and public bedroom qarters, where everybody sleep. Rich guys may build houses in distant places, trying to keep same body for long time.
Machines which can automatically take care of children and elders would improve situation too. As replication was never considered somewhat really private, when tech level reaches big hights making it possible to grow children from scratch without having sex, it will likely eleminate old way, especially if being supported by government.
Everyone in cities will be entrusted with keeping their current bodies within reasonable state to keep every body healthy. Health will likely be tracked every bed exit/enterance, noting body vandalism and writing out fines from such souls.
Medicine will be kept by government because bodies aren't private property at all. Ones who need long-time recovery could be moved into distant machine-operated hospitals as well.
Non-mental contests will likely never became popular or even existent — it will seem clear for everyone that bodies is somewhat unimportant in their lives.
Prisons will likely be distant machine-operated places as well — you don't wanna let criminals escape via soul exchange!
As for elders, it is hard to say to which morality that society will attach. It seem reasonable to make criminals take old bodies to let good people use healthy ones.
Overall, future doesn't seem to dark anyways. Body shuffle works nicely in two cases:
- Small society with absolute trust. This is also applicable to space expeditions in modern word.
- Big society with SoulIDs.
So answer depends on ability to get SoulID: if you can, huge cities with modern technologies is possible, if not, small towns still can survive quite happily. If it is possible to inhibit body shuffle, result depend on when they discover it: it will lead to catastrophy in medival times but can be used in quite good ways if discovered in modern world.