What’s better than Google’s grid icon for switching environments?

I'm working on a project for a client. The UI I'm designing has a header bar and in the header, there is a place where the user can switch environments; there are four. So for example I could be in the marketplace environment, then switch to one of the other environments from the header. All the environments live under one umbrella organisation.

The client is very attached to using the grid icon that Google uses to switch apps. (See attachment).

The site I'm designing is for public citizens; so there is a wide spectrum of people we're designing for.

My concern with Google's grid icon is that although it's a Google thing, not everyone is going to understand what it is or what it does, especially outside of the Google context.

I presented more or less a drop-down style approach so that it was obvious that "hey, this is a menu of things" but I'm getting pushback.

I'd be curious to hear any arguments for or against the grid icon and would love to hear if anyone has seen better examples of an icon or indication for a menu where the user can switch context completely.

Thanks everyone!