I am not sure what you are talking about or to whom you are talking to.

Itch does not "list" game engines. Or games, or anthing else on this site. Oh, they show the list, but they do not fill it.

So if you look here https://itch.io/tools/tag-game-engine , a thing like https://godotengine.itch.io/godot was listed by this user: https://godotengine.itch.io/ . The only "official" thing about this, is the fact that this account has the verified account checkmark. Something you see rarely on games.

The account owner listed the project under these tags in the tool category:

2D, 3D, free, Game engine, Godot, godot-engine, libre, Open Source

I especially do not understand what you mean with existing and not existing. Are you talking about meta information? If you cannot select a made-with option for an engine you used, use a regular tag, if you wanna brag how your game was made. Unfortunately, both mechanisms are used and not all engines are in https://itch.io/game-development/engines or the meta info.

For the games actually made with an engine, you can filter this way:

https://itch.io/games/tag-godot

https://itch.io/games/made-with-godot

And yes, some people will look for the information which tools were used to make a game. I know firsthand, because I do this occasionally. Doing so is a bit clumsy, because like I mentioned, the usage of the engine information is not unified. It could be a tag, it could be a meta-information, it could not be given at all.