That is correct.
However, this is a quasi-monopoly by google having quietly overwhelmed the space. Same thing for RCS messaging.
Neither push notifications nor RCS are proprietary, so there is a possibility to tear oneself from google here.
For instance, there are several free and paid push notifications services. Pushbullet is a popular paid one, not too expensive. I personally use https://ntfy.sh/, which can be self-hosted.
RCS is different because trusting the encryption keys makes RCS work, so there would have to be a critical mass of buy-in to use an alternative to google's RCS implementation.