Why is Family Guy such a popular meme format?

Its been on for a long enough time that you can find a scene suitable for any meme. Ditto the Simpsons.

Because it sucks. Kids who grew up watching it and thought it was the funniest shit ever are older now and they know it doesn't hold up so it just became meme fodder.

13 year olds. Basically, like bunhead pointed out, it's part of the post-ironic meme phase; it was popular with that demographic at the time and now we kind of look back with embarrassment (kind of like "this will be x in 2012)

I think you touched upon the reason without realizing it: it's popular in a lot of the deep fried / layers of irony memes that I see because its understood as a mediocre show for boomers.
Like, the animation has aged poorly and the humor is outdated, but the characters are recognizable. I think the humor of these kinds of memes comes from seeing the show's characters in a different context of humor than what they are normally in. Like, I can picture a meme of Brian holding a margarita glass with the bottom text being something like "doodoo ass." If you made a meme of Peter Griffin, running away from a picture of Chiang Kai-shek and saying something like "No Mao don't stop until you capture Taiwan," people here would probably think it's funny, not because it's related in any way to the show, but because it's disconnected from that show. Idk, I think that is at least an aspect of it