To be clear, if it was the same IP address, then it was the same server, not “both servers.”
Not necessarily, but probably. Both domains could have the same MX records, with the server for one domain acting as a relay for the server of the other domain. The easiest and most likely scenario is that they simply set up the server to handle the emails for both domains. It's entirely possible that any given mailbox on the server could have addresses from both domains assigned to it.