I wonder how to figure out whether one Bitcoin transaction is from the same wallet (public key) as any previous transaction in order to proof the ownership of a wallet.
If the transaction in question consumes inputs that were locked to a bitcoin address that has been previously used, then it would be easy to tell that the payment came from the same wallet. After all, the payment came 'from the same address' as a previous payment.
However, re-using addresses is not best practices. Doing so is very bad for your privacy, just consider the premise of this question to understand why! There is no reason to re-use addresses and thus allow someone to see your financial history (ie, all of the other transactions you've made from that address).
Thankfully, modern wallets are built to never re-use addresses. Further, addresses from the same wallet are indistinguishable from random addresses, there is no public information included in the address/transaction that links them together. This is by design: you wouldn't want someone to be able to determine your entire wallet balance, after receiving a single transaction from you.