Same problem here. Weirdly, Firefox works fine on my laptop which I constantly shut down/reboot throughout the day. But on my desktop PC, it throws me out of almost every account as soon as I close Firefox. I've checked the settings and it's not on "delete cookies upon closing Firefox". so I don't understand why it's a hit-and-miss on two computers with synced add-ons and everything.
This started happening with the latest update to Beta 118.0. It may be bad enough to throw me off the Beta track, but I'll wait a few days for the problem to be fixed before downgrading to FF 117.0.1. Meanwhile, I can use Safari with no issue, which would not be a permanent solution.