A shared or hashed secret would do it.
Plenty of exchanges don't know their customers, and in fact that is how they get their customers.
No. Coinbase deals with fiat money, therefore subject to AML and KYC regulations.
The question was about customer support. AML and KYC regulations do not require that customer support persons know your PII. That can be kept firewalled from them.
That's not related to customer support, though. It's more like customer surveillance.