I also have a problem with labels but a different one. When I have labelled my files and reopen Bridge again later the labels are gone. And the illustrator files have a Bitmap color mode. I have worked with Bridge CS6 for many years and thought it was a great programme but since I had to buy the 2021 version the problems started. And no one in this Adobe support community seems to know what the problem is...
So you are not the only one I'm afraid.
AdChoices