Most websites impose a lifespan on the logon credentials stored in your browser as a security feature. On FamilySearch, it is clearly 7 days, as they tell you.
Other sites are not so clear, and some sites invalidate lots of other remembered stuff. The National Library of Scotland maps site seems to forget my preferences around once a month, even though there is no username stored.
Ancestry and FindMyPast both often seem to demand a fresh copy of my credentials in the middle of a session - often between getting search results and wanting to see the actual record.
These things are stored in Cookies, so if you clear those, the website will ask afresh. I am selective about clearing the cookies on my browser, and I still get hit.
One thing is for sure - the sites will demand logins when it is least convenient for you, and when your time is limited.
