Here's my take on it: the interview is testing your ability to "show" that you absorbed the content and process of your 4th year data-structures and algorithms classes. The whiteboard process is all about showing them that you can regurgitate what those courses taught you, likely in the manner that the prof taught you. You're essentially playing "professor" for them in front of a "class", for a short period of time.
Not showing them that you can write
code. The code you write on the whiteboard will look nothing like anything you'll ever write at Google.
I think one way to do well in that interview, is to pretend you're the professor and they're a student, and you're working through course material.
In the end, it's just showing them that you passed the socio-cultural hurdles they think are necessary, even if they no longer explicitly check GPAs and which school you came from at interview time.
Why?
Google is more often than not the first job these employees have after school, and many stay there almost forever after. A large percentage are masters or PhDs.
Google is founded by two Stanford grads, both children of academics, who never worked in the industry outside of Google.
Academia is hence the reference point against which Google measures things.
Google is structured in many ways just like a university. Publish or perish (Design docs, PRDs). Thesis committees (perf "calibration" committee, interview committees, etc) and review (intense code review). Even down to the physical "campus" structure. On site cafeterias, even housing/dorms (GSuites, etc)
It's something that was very foreign to me having worked in the industry for a decade before, without a degree.