Why isn't there an option to not allow my username to appear on Google's search engine?

Because it's Gelbooru software, the guy who coded it didn't think anyone would use a sensitive name on a porn site...

There is no way for us to remove it or change it etc, because the software is basic as you can get and we are under copyright so we can't change it either.

Besides Google is the one scanning the site and listing the user names, not us.