I’m pretty sure this is all about convenience. The micro sign is supposed to be the Greek letter mu, so they should be interchangeable anyway. The character is probably in Unicode twice because it has been added from both perspectives (We should have the Greek alphabet! We should have mathematical symbols!). Whoever updates that website just uses the character that is easiest to type (my keyboard at home has a mu, probably the micro sign) oblivious of the different encoding.
MusicBrainz does choose one encoding to use for all occurrences of some symbols. There was some discussion recently to store all ‘real’ hyphens as hyphen-minus, because they are usually represented with the same symbol in fonts anyway. MusicBrainz also seems to store half spaces as normal spaces, which is annoying because there is a big difference between the two.