While using Safari, I am being prompted to update Safari when system preferences says system is updated?

Catalina is no longer up to date with Safari. You need Big Sur to get a more current Safari. Other browsers are more current under Catalina if your Mac can't be upgraded further.

Browsers by Mac operating system - Apple Community

those messages are generally generated by the set visited. Incidentally, upgrading Safari will not always stop those messages.

I prefer Firefox as an "update" to an old Safari.