Schwab Bank is indeed a real bank. That's why you can write and clear checks on your Schwab Bank account. No doubt Schwab maintains correspondent banking relationships with large banks in major financial centers, like JP Morgan. It's very common for smaller banks to establish these relationships, especially for clearing international transactions.
Schwab Brokerage is a separate legal entity and not a bank, though Schwab in fact operates pretty seamlessly between the bank and the brokerage so they do seem much as one.