Your identity is anything you define yourself with. You just identified yourself as a picky eater -- that's part of your identity. However, you choose not to make it a celebrated, core, or significant part of your identity. That's fine.

If you said "I am not a picky eater ", that's denying it's part of your identity.

Ones identity is simply the set of all traits they would use to self describe. Parts of one's identity people will push forward and parts they'll not ever emphasize or show.