LinkedIn too behaves strangely on Firefox/Ubuntu. Endless login loop, even if all add-ons are disabled. I'm surprised that such popular websites treat Firefox so poorly.
Fellow Firefox/Ubuntu user here. Are you blocking Captchas somehow? This is usually my problem (and specifically very recently with LinkedIn, hence the comment).
I don't think I have changed anything related to captchas, at least not intentionally.
I see a lot of "Request to access cookies or storage on https://www.linkedin.com/ was blocked because of custom cookie permission." in console.
However, my Preferences > Content Blocking is set to "Standard"; no add-ons are enabled; not a private window. I have DNS-over-HTTPS enabled. There are no linkedin-related entries in about:config.
Have you got a Pi-Hole setup or some such? LinkedIn on FF/Ubuntu works fine for me at home, but I think I had to whitelist something on Pi-Hole admin console.
Since the console errors were about cookies, I checked under Cookie and Site Data. There are no disabled domains. LinkedIn domains show a few KBs of storage, which I assume means LinkedIn cookies are not being blocked.