No, something being under github.com/google means the person who started it was paid by Google, not paid by Google to code this. Google contracts (like most tech contracts in the US) have ridiculously broad IP assignment clauses, so unless you go through a lengthy process to request Google disown something, they own anything you code, and they insist you open source your things under github.com/google.

You decide your own definitions, but that's very different from "Gmail by Google" or even "Go by Google" in my book. Note how the main author has "Ex-Google" in their bio, too.

Not necessarily. You can write open source code in your own time and publish under Google org on GitHub. This is the recommended process if you don’t care about retaining the copyright to your code.

If someone does want to retain copyright, there’s another process for getting approval.

They do, otherwise it won’t be in that repo