Yes, it is fools mate but it is not useful because no one comes in this kind of mate. (rarely 1 or 2 time)
Fools mate

I'm not sure what is most common, but here is a real example from one of my early games at chess.com. It isn't a perfect 2 turn loss, I lost on turn 4 but the checkmate pattern is the same. Don't try this at home. =)
This is actually a useful idea to know. I was playing a game and I blundered a piece. Then I saw a swindle, and I went for it. I sacc'd two pieces, and when he took the second (he would have been fine if he hadn't), he was done. I made a few forcing moves then dropped my queen on a square, forcing him to take it. When he did so, he covered up all the squares except the f7 and my bishop came out for the fool's mate variant. It was pretty epic. Saying you checkmated someone with just a bishop is probably even rarer than with a knight.
Cool. Could you post the game here for analysis by some cool analysis people?
Assuming you meant me, I don't have it. I can try to devise a similar problem, though. Give me a minute...
Here, I contrived something. I believe this is a little more forced than what happened in my game . . . I think this is a legal position.
Here's a fool's mate variation I played a moment ago in a blitz game.
Don't mind the opening, but that's a legitimate game I just played.
how can you fools mate? what is the most common way?
i think its something like this.