Yes, you are right, they receive from that company wallet ID, and other creditials. They tell me also receive notifications from that company about BTC price changes.
It's closer to a mutual fund company in terms of how to manually tell customers the latest bitcoin prices and while they don't let you have btc in a personal wallet.
Sorry to say, I'm not even sure that any bank can reach the transaction history of 11 years ago. So the only clue is the company email sending you the latest bitcoin price notifications(if you keep it).