Nah, I don't think 'spy' should be a genre. It's more of an activity (that happens in many kinds of stories) than a type of story in itself. I mean, you could write a spy romance, spy horror, spy comedy, or the classic spy action mystery.
If anything, I'd consider it a sub-genre of Mystery, along with Detective Noir and Stranger Things-type stuff (y'know, "kids on bikes").