Finally got the museum to work.
However, nothing makes really sense. It says that I own 360 on the card but apparently the Series console is my first Xbox hardware....
Also I got 0 achievements even though my first rare achievements seems to be a recent one from the San Andreas Definitive Edition. My most played game shows a picture of AC Odyssey (which is correct) but the text says CoD BlOps II (eew).
At least the summary card seems to be a little better:
Gamerscore is correct but I'm missing almost 10,000 achievements and over 200 played games.