EDIT: I just fixed it, but I don't really know how. I only did similar things I tried before and after clicking on a correct link, the family in my account got replaced with "Family 3" (or some other preset name) where I'm not the founder. Thanks for helping! :)
I also assumed that it has to do something with the link, but I was there when he shared the family with me, and he definitely clicked on my bubble in the tree to e-mail me the link. That's why I tried sharing the link again to myself and to different e-mail addresses, and I tried removing the family from my account and joining again via the link, but always with the same outcome. I don't know what else I could try.
I'm afraid I have to create a new account to fix this…