Is my phone battery genuine?

You can check if your battery is genuine here:

Settings → General → About

This feature works on iPhone XS and newer models, with iOS 15.2 and up.

You can find more information here:

About genuine iPhone batteries - Apple Support

Did you have it replaced? Was it replaced by Apple or an authorized service provider?

If it was replaced by someone other than Apple or an authorized service provider, then no. It is not "genuine".