How can I tell if my iPhone battery is genuine?

If its detected to not be genuine, it will issue a message stating as much.

link ➡️ About genuine iPhone batteries - Apple Support

If you had it done at Apple or authorized service provider - yes.

If done elsewhere, nobody knows except the person who bought and installed it.

If its not showing any message, then its a genuine Apple part. Or at least not identified as a fake one.

If it's not charging, take it back to where the battery was replaced, and have them take a look at it again.

Yes, you can see whether the part is genuine or not in the service history:

iPhone parts and service history - Apple Support

Who changed the battery?

If you did it yourself, your phone is unsupportable.