Why did laravel stop setting cookies and issue Token mismatch?

Question

Laravel

I set up ajax post requests with the transfer of a token to OpenServer and Laravel "broke". Now throws a Token mismatch exception on any forms, even on the standard login and register, although the Token is passed correctly in the code.
In the Network tab in Chrome Dev Tools, you can see that no cookie is set in the browser, although sessions are written in framework / session, that is, with write permissions to the framework / session folder, everything is fine.

On the other neighboring Laravel, everything is OK, the cookie is set. The configs were slightly different, I copied the config from that Laravel where everything is OK (the cookie is set) to the one where it is not set, but it did not work. Googled a lot of everything related to Token Mismatch, many people have a problem, but have not yet found a clear solution.
Maybe someone came across? How to fix it?