The rails app can also store a cookie (for example use the rails session cookie) in the clients cookie store. This gets sent to the rails app on every request by the react app and you can verify the identity of the user by reading this cookie.
There are very few adjustments needed for this approach (and they would also be required for JWT - which I don't want to badmouth, it is also a viable and good approach).
An upshot of this is that almost no data needs to be stored at the client. Only an ID that makes it easy for the rails backend to identify the user and achieve all needed informations.
On the other side, if you want to store all the data at client side this is also possible.
I would say there is no clear advantage of rails session over JWT or vice versa. But generally this is how it should be implemented: By giving the client the necessary information which then gets passed over to the backend again.