I'm using a HTTP server, so for sure Godot don't keep $COOKIE and $SESSION as you mentioned because it isn't a browser (but would be great if it could keep cookies and session stuff like a browser while connecting). The problem with a file on the user's PC is that you could actually move this file to other computer and access someone else's account, I'm trying to avoid this. If only on Desktop there was a PC Unique ID, this could be achieved.
I kinda solved it by adding a database sessions table, when you tap Remember Me, I do a hash of user's email+extra-encrypted password and store at user computer and send it together with a hash of some IP info, and store it together with the userID given by email+pass, so when I try to login, I check the file hash with the email+pass hash from server, together with the user IP info hash, with this, if the user changes it's password, it don't auto log in.