> how do you proof that you send someone funds (in LN)?

Typically, the recipient gives the sender a receipt, and cannot sweep the funds from the LN htlc without doing so. Notably, there is an exception: it's possible (and common) to set up a payment that does not require a receipt from the recipient. If you do that then you simply do not get a proof of payment.

> [In monero] you can't proof who owns [an] address [or] if that money actually gets used/moved

There are four techniques outlined on moneroleaks.xyz for finding out who owns an address and/or if the money sent to that address actually got used/moved. These techniques have all been used to trace and convict monero users. I won't let monero bros pretend to have any intellectual honesty while they continually ignore widely known and documented circumstances while insisting they are impossible.