An account ownership check is another name for account ownership verification, and that entry covers the subject in full. The phrase describes the single step of confirming that a bank account really belongs to the person or business that claims it, before a payment, payout or direct debit is set up.
Firms run the check in several ways. They may compare the name given by the customer with the name the bank holds, using services such as Confirmation of Payee; read the account holder's name through an account information API with the customer's consent; or send small test deposits that the customer must confirm. The aim is to stop payouts to the wrong account and to block fraudsters who try to link someone else's account.
In everyday use, "check" tends to mean one automated look-up at a point in time, while "verification" can describe the wider process, including documents and follow-up reviews. Both sit within bank account verification and broader know your customer controls.