It does seem to be notifications on the phone, but (a) that's incredibly surprising and disturbing and (b) it's really unclear why or how that would work when a phone is disconnected from the network. In any event, Google inserting themselves into notifications would be tantamount to reading all my email, texts and everything else, so ... why wouldn't this be restricted to opt-in? Many questions.

The article says that Google and Apple know about the push notifications being shown on the phone and governments can make these companies turn over customer data.

I'm not sure if it only covers (for example) the unified notification service on Android or whether Apple and Google know of notifications that don't make use of that API. It's not clear from the article.

I don't know about Apple but on Android it's almost a capital sin to strive to use other services, and they work a lot worse than GCM (because of all the artificial limitations that Google imposed over the years).

A push notification is generally what creates the "you have a new message on <app>" red bubble.