others have mentioned microG, but if that does not answer your question it may help to concretize what services you need from the google services stack. this page has a table on what services did the microG project implement so far, it may help in deciding on what exact services do you need

for push notifications I recommend ntfy or another UnifiedPush-based solution, but be aware that ntfy's official instance but probably the mozilla instance too is being hosted in the USA. if that's a problem, the ntfy app let's you set a custom ntfy server (find a public one or host it yourself), and all apps should handle it without difficulties which register at the Ntfy app after you made this change.
with UnifiedPush, there's no central point of failure like with google firebase, all services get told where to send the notifications, and that is determined by the "UnifiedPush Distributor" app on your phone (of which the Ntfy app is an implementation, along with SunUp and NextPush, they are also distributors)

You need to run your own push services provider like ntfy or gotify and tie all your notifications into it

Look into /e/OS , lineage, and graphene. On the /e/os page, there is a big explanation also on grapahenes documentation page. That helped me a lot.

F-Droid and Aurora exist, i use them on my pixel 3a xl running ubuntu touch/waydroid.

A small addition to already mentioned stuff. There are multiple ways to deliver the notifications without google services. 1st and older one is by simply letting the app hang in the background indefinitely and ping the necessary servers from time to time, that one almost always works, since app developers can't really rely on gapps being installed; 2nd is UnifiedPush (that's already mentioned sunup [mozilla], but also ntfy [ntfy], nextpush [nextcloud], gCompat-UP [google firebase], NoProvider2Push [fully local]). AFAIK, it works similarly to the way gapps send notifications and uses less battery, but not all apps support it, so you may need to search for forks. For example, the official and, iirc, Foss telegram clients don't, but mercury, nagram{,x} and momo do.

Its one or the other. Either Google Play Services will push notifications, or the apps have to have the ability to handle push notifications on their own (which isn't common).

Google Play Services can be sandboxed in GrapheneOS, but there isn't an open source Google Play Services since its not included in AOSP. It is very much a proprietary blob.

As other comments mentioned, Push Notifications, your main issue has workarounds. So, the answer would be Yes.

BUT Google Play Services is much more than that. This dependency can't be worked around really and it's one more way that Google establishes his stance in this oligopoly. I'm certain Apple has something similar if not more aggressive. This is the reason why the year of the Linux Phone is so far away.

Personally, I think Linux Phones will catch-up when their hardware allows for emulation of an android subsystem where we could sideload (illegally?) Google Play Services...

An app? Nope. For notifications, there is open source alternatives to Google and Apple services but it is used in the apps side, not users side. Have a look on microG and Open GApps to flash in your Android device; it might help you.

I'm not sure what you're asking. There is microG that spoofs Play services to apps. That's been around for several years.

You can also find open source push notification apps on f-droid. Some of those might require a self hosted server, while for example Sunup uses Mozilla's server.

Whether those work on your device probably depends on your ability to root it 🤷

Nope