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.
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.