Can I copyright code or an app?

Yes. Copyright can cover a variety of things, including text, drawings, photographs, and code (including both source code and object code).

Recent rulings by the Copyright Office have said that AI-generated works are not copyrightable, so you might have to have written the code yourself. It is even possible to have a copyright in code that you maintain as a trade secret, so long as (a) you actually maintain the code as a secret. See the discussion of what is needed for something to be a trade secret.