That would make sense if apps like Twitter or Reddit were banned from the app store or had also decided to ban NSFW content, but they seem to be getting along fine with their verification-free "click here if you're an adult" systems. Why was Tumblr different?

To be sure, this move by Discord seems like a reflection of the same kind of decision that Tumblr made. But blaming it on the app store in Tumblr's case doesn't seem to make sense. Discord seems to be doing it either to comply with increasingly fragmented laws, out of actual moral desire to fix a problem, or to avoid lawsuits-- not because of app store policies.