Do you know if I run tagging in Musicbee if it writes the tags to the actual files
or if it just keeps it in a database in Musicbee?
Regular tags are saved in the files.
Custom tags you can decide for yourself.
Statistics such as playcount in the library.
You can use the Tag Editor to see what tags are actually in the file.
Also note that if tagging is of essential importance, the general opinion and recommendation here is not to use wav but flac.