I don't know how Signal works and I never used it, but could I signup with a phone number and keep using it with another number, on the same phone?

Yes. The phone number is just for activation, once activated, you can swap the SIM and carry on. Or have the SIM that receives the activation text in another phone, or be virtual, or whatever.

I doubt they are discarded when push notifications exist

push notifications are not related to phone number, but rather to a randomly generated token in app.

Another comment contradicted this.[1]

[1] https://news.ycombinator.com/item?id=46959019