How quickly or how often does it happen? Is it related to exiting/quitting Firefox, or a certain amount of time elapsed since you were active on the site, or... no pattern?

Generally speaking, you are able to stay signed in because the site set a "persistent" cookie and nothing clears the cookie. Persistent just means that it has a specific future expiration date, and isn't a Session cookie meant to expire when you Exit/Quit Firefox. Although Firefox sometimes lumps together cache, cookies, and local storage, the cookies are the most critical for this problem.

The pattern seems random. Usually, I get logged out when I open Firefox after updating it, but sometimes I'll randomly get logged out of one or all websites after a random period of time.

We need to know what sites these are to be certain what is happening or why it's doing that. Some sites like Banking and Medical sites and Gov. aka SS, Medicare will log you out if it detects inactive after specific time as well for security purpose and data privacy. Well updating would itself log you out because it has to restart the software.

Just any sites in general, it's not exclusive to sites like the ones the answerer mentioned. For example, Fandom/Wikia.