Does setting requireMatchingUserAgentForSession to false have any additional implications to be aware of?
The reason Craft stores the user agent string in the authentication cookie when logging in is to help prevent session/cookie hijacking.  If will validate that it is the same subsequent requests on the presumption if your cookie is hijacked, there's a significant chance that the browser that stole the cookie will have a different user agent string that the one the session was started with.

Granted, this isn't fool proof (user agent strings are easily spoofed) and some browsers store their version number (like you discovered) in the user agent string as well.  This is all well-and-good until your browser decides to auto-update, forcing the string to change and then auto-logging you out of Craft, which looks like what you're seeing.