I'd imagine that for a given underlying foreign security, it's possible for there to be multiple U.S.-based banks, each of which acts as a custodian/depositary for ADRs of that foreign security. In which case, when you buy an ADR, does the ADR you buy have a particular custodian associated with it? And are ADRs with the same underlying security all traded via the same book even if they have different custodians?
Yes, it is possible for more than one bank to decide to sponsor an ADR on the same security. In this case the ADRs have different tickers and are conceptually different securities. An example is the Japanese company Daikin (Tokyo 6367) which trades in US as both DKILY and DKILF (the latter is much less traded however, typically one ADR prevails over the other in the marketplace, in tems of liqidity).
DKILY is with Citibank, DKILF is with Bank of New York Mellon. So that is how you know who the custodian is, from the ticker.
ADRs can be issued by any depository bank and if there's demand multiple banks can issue an ADR on the same company. In case of the multinational banks like BAML, CITY, BNYM, WFC, DB they are usually also the custodian as they have a presence in the home market of the underlying security.
In case of no presence in the home market of the underlying security, they have to arrange for a custodian. This however adds a higher fee burden, making the ADR less attractive for investors, hence not very often done.
Also have a look at the SEC, Investor Bulletin: American Depositary Receipts