How to use (or should I use) time code/user bit feature on Sony FDR-X3000 action cam
No, they don't have a way to keep it in sync.  This is why the timecode is clock based.  The idea is that if the clocks are reasonably close, you'll be able to line up multiple cameras across multiple shots once and note their offset.  You can then use this as a starting point on other shots that are done at the same time.  Clock drift is likely to require minor corrections though, but it can at least get you close.

In professional settings, there is a system called genlock where the frame timing is based on an external clock signal that keeps all cameras exactly in sync and it's even possible to provide an external timecode such that all cameras have exactly matching timecode that doesn't have to be based on real time, but can be based on shooting time instead.

The user bit information is for capturing other additional data about the shot that isn't caught in the timecode.  Date isn't generally included in timecode and scene number certainly isn't.  This meta data is helpful for sorting out clips in post production and is basically a digital version of the information seen on a slate at the beginning of production shots.

More advanced cameras support additional meta-data.  My Ursa Mini for example can record the name of the project, director, camera operator, various lens and shooting conditions, scene number, reel number, take number, whether the take was good or not, if it was shot inside or outside, day or night, etc.