I eventually contacted Microsoft. What I had to do was to stay signed in with my microsoft credentials, on e.g. Word, on the old PC. Then start e.g. Word (actually any office app) on my new PC (office already preinstalled), and then sign in there as well, with my MS credentials. Word automatically presented me a window showing the licensed Office Home and Business 2021 which I have on my old PC. I just had to select that, and Office is now available also on my new PC. Interestingly, at the moment, I can use Office on both my PCs (old and new), but I guess this is only temporary. Will uninstall Office from my old PC anyway now.