I have a similar issue (also Firefox on [K]ubuntu 22.04) every time I open a link on a logged-in site in a new tab, but in my case merely refreshing the page is enough to get me logged back in.
I assume is most likely the fault of the fairly aggressive mix of extensions I’m running rather than Firefox itself, but I haven’t actually tried to troubleshoot it yet.
Not for long if Lennart has anything to say about it, I’m sure.