Auto register new site?

Scenario: I have a self hosted PiWik

Is there a piece of code I can put on my new site that will “phone home” and have that site show up in my Dashboard?

I install a bunch of sites daily and would like to speed this up.

Bonus question: is there something I can do to make each new site I install unique so as to avoid a footprint in Google?

Sure, but you need to use the API to create new websites automatically, and then insert the JS code in your website pages. See a similar use case here: Log Analytics: a Business Use Case - Analytics Platform - Matomo

For anonymisation in google, see: GitHub - matomo-org/tracker-proxy: HTTP proxy for Piwik's tracker API. This script allows to track websites with Piwik without revealing to your visitors the secret Piwik server URL.