Can I back up my Apple watch on my Apple Watch?

If you backed up your old phone you and restore the new phone from that backup and an old watch backup may then show and be available when you pair the new phone to the watch.

To pair the watch to the new phone you will first have to erase it. Se the topic:

If you don't have your old iPhone or you erased it

in How to pair your Apple Watch with a new iPhone - Apple Support

No,your backups are either in icloud or you can make a backup on a laptop/computer with itunes.