Why would anyone need/want a third party wallet?

I noticed that some sellers/traders that accept ETH require the customer to have his ETH wallet with a thirty party, from the list of third parties they support. Why is that? The way I see it all they would need from the customer is his ETH account number to which they could deposit funds or wait for the customer to send a payment to them from that account. What ETH account management software (that is, the wallet) the customer uses should be irrelevant to them.

Any ideas? Thanks