With Public Key cryptography, which is used in Blockchains and other Distributed Ledger Technologies (DLT), the ownership of the assets is directly linked to the correspondent Private Key. In order to prove ownership of these crypto currencies you need to prove that you know the respective Private Key. However, revealing the Private Key itself would be inappropriate due to a number of reasons. The most critical being that you compromise the security of your crypto currencies (the counterparty knowing your Private Key can do everything with your crypto currencies).
Therefore, it doesn’t make sense to share the Private Key directly in order to prove ownership. Some companies prove ownership by transferring a small amount from the respective address to the address of the third-party that wants to check ownership. However, also this method is prone to fraud and not reliable at all. For example, I could instruct a crypto exchange to transfer some amount to a third-party address. If that crypto exchange transfers this amount from an omnibus wallet the third-party could wrongly assume that I have ownership of all the funds on the respective address although I obviously don’t have.
The most reliable way to prove ownership of crypto currencies is to sign a specified message with your Private Key. By doing so, the third-party can verify that the counterparty really knows the respective Private Key without the need of revealing the very key or having to send a transaction.