So AIM was built on an existing chat protocol called OSCAR. The same protocol used in other services. So people eventually figured out how to make chat clients that could log into many different IM services on one app.
This was not sanctioned by AOL, but they allowed it at first. Then they decided you HAVE to use the official AIM client to talk to people on AIM. The third-party developers ignored AOL, so they entered into a tug-a-war match for a while.
Because AOL was using known software to make AIM work, there was only so much they could do to keep their client working while also blocking everyone else. Eventually it became too much of a hassle, so AOL relented and third-party clients kept working until the service was shutdown.