I'm not going to look at the New York state rules of civil procedure to confirm specifically, but no. Document discovery gives a party to a proceeding the right to access documents in the other party's possession relevant to a material fact at issue in a trial. It is not a pass to demand anything you want from them.

Without knowing the specifics, I doubt Drake's complete text message history is relevant to any material issue on whether UMG conspired with Spotify to boost Kendrick streams.