How would that work? The provider / vendor of whatever app or service you are using would need to know that they need to contact that server to send a notification to your device.
Apps on android register push notifications with the Google Play Services daemon on your phone, which in turn registers them on Google Cloud Messaging. Google Play Services is the non-open source part of Android that makes Android a proprietary operating system.
MicroG is an open-source reimplementation of Google Play Services, used on Android variants like CalyxOS and Lineage OS. MicroG currently only supports registering push notifications on Google Cloud Messaging, but I would love to have an option for self-hosted push notifications.
Right now looking at my MicroG, I have 77 apps registered to use Google Cloud Messaging.
One benefit of using MicroG at least, is that Google doesn't know who owns the phone getting the push notifications. There is no Google acount or any other identifying data, the IMEI, serial number, etc is randomized and fake.