I own physical copies of both Tasha’s and Zanathar’s sourcebooks, but I don’t know how to let the site know that. Is there a way to register physical copies?

There is not. Physical and Digital purchases are separate products.

They do offer a bundled of both physical and digital when purchased together in the Marketplace on DNDBeyond however for future reference.

So my book collection is worthless! If I want to use this site or the app I have to buy them all a second time! 

you can also homebrew everything but clases