Hi @felix0712 and welcome to the community! 🙂 Unfortunately, you would need to start over to have the account registered to you instead. I have copied over Spotify's FAQ on how to do this from the Family support page. Hope it helps and have a good one! 🙂
Change the owner
It's not possible to change the owner of your Premium for Family plan. The current owner would first need tocancel. Then, when all accounts revert to the free, ad-supported service at the next billing date, a new owner can start a new plan.