Bridge does not change meta data (e.g. stars) in file meta data

Most likely Bridge is updating the embedded rating metadata correctly. The problem is that there are multiple tags where rating can be saved in a file and Windows and Bridge don't consistently use the same one(s) for. Video and audio files seem to be very inconsistent, image files work a little better. Yes, this is annoying.

Bridge uses xmp:Rating which is preferred because XMP is an open standard and is supported by most photo applications. Windows uses their own rating properties which almost no other applications use. I'm not sure, but I think Mac OS has its own rating tag also.

For rating, Windows uses a tag called "Shared User Rating" while XMP uses <xmp:Rating>
Just to make it even more confusing, Windows uses <MicrosoftPhoto_1_:Rating> for image files.

I don't think there is a solution for seeing ratings in both Bridge and File Explorer. This conversation on the ExifTool forum explains the situation a bit more. It looks like ExifTool can write to "Shared User Rating", but that would add a lot of steps to your workflow and doesn't seem like a solution.

I would recommend using Bridge ratings since they are more likely to be usable in other applications. 

I work on Windows 10 and use Bridge 2023.