I think you're conflating something here. Whether you send a session ID in a cookie or a JWT makes no difference for the app's general behavior, even when you use multiple tabs or multiple devices.

But I remember a time when especially bank websites added an additional token (like a super strict CSRF token) to their app, which tracked the current page of a user and if you browsed the same website from another tab, this other tab didn't have the proper token and the whole thing just returned "invalid user action" or something like that.

However, this has nothing to do with session cookies.