@sveakul: Not important for the problem / just explanations to your questions
OK, as I already told you, did not looked at the file extension/name when I answered and just supposed it was that kind of file (again sorry, my fault)... In any case, this has nothing to do with the problem I depicted in my first post, and frankly has 0 relevance to solve it.
Answering what I don't like about the portable one: I don't like having programs installed in "non standard" folders, I am an industrial programmer, use my computer for work and try to keep everything as well organized as possible otherwise it gets messy in no time. Apart of that I have no problems with the portable installation except it doesn't work like the other ones.
What I love about MusicBee is the simple interface, hability to read FLAC files, that it remembers the file it was playing when closing it. I don't need anything else... backing it up therefore is not important to me as a clean installation just works as I want.
Now, as this is the bug reporting forum, I'd like to keep this post as focused on the real issue as possible:
Following with the real problem, which now is much more tested and I can explain even better:
After all the suggested tests, which included uninstalling the Microsoft store version, installing the normal version and then uninstalling it to install the portable version, all of them suffer from the same problem in the latest release: media keys are not working properly.
Installing an older version gets media keys working again, so for any reason there is something wrong in the latest version which made all this stop working properly.
Thank you all for your help!