I have seen some PS battles on YouTube, but never monetized. If they are monetized, then maybe it was Nintendo doing their knee-jerk reaction to take money from monetizing YouTubers.
But even ignoring that, if there is a YouTube video of a PS battle and it is monetized for the user, then that's illegal. Unless you live in a place where copyright laws don't exist, like Morocco or Sweden or something.
However, just because it is illegal doesn't mean that you can't do it. You can monetize Pokémon Showdown videos. But by doing so, you allow either Zarel or Game Freak to come after you for doing so and they can take legal action against you.
Given that Game Freak isn't too hung up over people monetizing Pokémon-related videos on YouTube, I doubt that they will come after you, so you can do it without worry. But this is not a guarantee. They can still choose to come after you at any given time that they want. Zarel can do the same. I don't know how hung up he is on copyright infringement though.
Avatar made by User of Shadows.
From The Smog, Issue 32: "Now there is no excuse for anyone to post discussions on the PS! Forums nobody cares nor reads about."
Yeah, it's not like that's incredibly rude and offensive or anything... >_<