Ownership of a digital asset on a blockchain is shown through the possession of a unique private key that corresponds to a specific public address on the blockchain. The private key allows for control and access to the digital asset stored in the corresponding public address. The blockchain then maintains a ledger of all transactions related to the digital asset, making it transparent and tamper-proof. This allows for secure and verifiable transfer of ownership of the digital asset from one person to another through digital signatures created using the private key.