My understanding is it's because XMPP expects the client to be able to maintain a connection to the server, which is typically not possible on iOS. Apps on iOS are generally not permitted to run in the background, with a few limited exceptions.

Yes, exactly. And while currently there are ways to maintain a connection to a server on Android, it is clear that the platform is going to restrict this capability even further. Even now, if you run a background process, and start doing some memory-intensive operation, like taking a picture, your app is unceremoniously offloaded from memory, making you unable to receive messages, until the service permits a restart.

But one could open and close a desktop XMPP app as well at arbitrary times. What is the difference?

The difference is control. A person knows that if he disables a desktop client, he won't be receiving any messages.

A phone user expects that messages would be displayed to him in notifications when received. One does not expect to be cut off communications because he had pressed a 'home' button, but it's what happens, and a user does not have any control to prevent it from happening.