OK, I saw this headline elsewhere, and I said "this is stupid, I don't care". Then I read the article you posted, and through 90% of it I thought, "PETA is stupid, I don't care".
But the end of the article gets to the origin of this whole thing. Wikipedia posted the picture, and the photographer wanted it taken down (because he was releasing a book that included the picture). Wikipedia refused, saying that the monkey took a selfie, so it's not the photographer's property - an asinine and meritless claim by Wikipedia, in my opinion.
Now PETA steps in, but they're actually on the same side as the photographer. So although PETA makes a wacky claim that the monkey has rights - which is asinine and meritless - they're only doing so to help the photographer get the picture taken down by Wikipedia.
But they lost the case, so the photographer doesn't get the rights to the picture back from Wikipedia. I agree with the ruling, but I disagree with Wikipedia. The picture belongs to the photographer; it's his camera, and he set-up the picture. An animal has rights, but not on Wikipedia.