This protocol is XMPP in 99% of the time, both Apple and Google use XMPP to connect the phones to their push notification service.
On Android, some Matrix clients offer the option to use non-Google push notification channel via UnifiedPush. UnifiedPush needs a distributor, a service that runs on the phone and manages the delivery of push notifications. Conversations, a popular Android XMPP messenger, can be used as such distributor.
So if you want digital independence with Matrix and thus do not want to use Apple/Google push, you might end up installing an XMPP messenger, so that your Matrix messenger actually can show notifications. Matrix alone just doesn't provide what is needed for digital independence.