There’s no problem at all doing this. If you try to run it on both at the same time it will ask you if you want to kick out the first one because you can only have one user logged into the game at once.
There’s currently no ability to configure different user profiles, so it will forcibly sync your logbook and all your settings between every computer that you log in with the same XBOX user name. Great if it’s you on both computers, less so if it’s a different person in the household.
And yes, just go into the store and look at the stuff you own and install it again. I think you can install on something like ten computers, but again it will only let you run it on one at a time.
A question I have is whether I can purchase an actual license on the same XBOX account that I have Game Pass on, and then use both the Game Pass version and the purchased version on two computers simultaneously. I’m going to guess that this is impossible.