Welcome to Android Central! I don't think it's possible, probably because it's an essential app. I checked my Pixel 6 Pro, and I can't turn off Phone app notifications there either. What notifications do you want to silence or get rid of?
It first, I thought you were speaking specifically about the Phone app notifications - the actual ringtones and turning the screen on to let you know someone is calling. But then you say it's not a specific app, so I'm just trying to understand.
Have you tried changing the sound profile to "Silent"? There are usually 3 default sound profiles - all sounds on, vibrate only, and silent. One way to cycle through them is by tapping the speaker icon when you pull down the notification shade from the top.
That won't change the on-screen notifications though. That has to be set per app.