Nope, it's definitely trademark law. Here's Merriam-Webster's definition:
to use the Google search engine to obtain information about (as a person)
on the World Wide Web
The reason Google is alright with this is because it specifies that "to Google" is to use Google. That prevents the trademark from being diluted.
Similarly, Google wanted the council to declare ogooglebar as being unable to be found on Google, rather than just being unable to be found using any generic search engine.
This is a very important distinction. I imagine that if Merriam-Webster tried to define "to Google" as to use any generic search engine, Google would have lodged a similar complaint.