Almost all media players use the same strategy. As it is very time consuming to read tags from audio files, they simply scan them all and store the tags in a database. Querying a database is far more efficient than scanning all those files. A logical consequence is that the content of the tags in the audio files and the values in the database should remain equal. If you tag inside Musicbee, this is easy to solve. Write the new values to both database and file.
If you use a third party tool like MP3Tag, this won't work as MP3Tag write tags only.
So you must use Musicbee to sync.
This can be done by Edit > Preferences > Library. Configure Musicbee to monitor the folders at startup or continuously.
Likewise if you have doubts about the values in a track, Select the track(s) , Right Click > Send to > File rescan