I think everybody has something about them they don't like being made fun of for. Most of the people who say "C'mon! Can't you take a joke" would probably react the same way if there was a joke directed at them about something they were insecure about.
I kind of wonder if this guy gets singled out a lot or if there's something else going on as well and this was a breaking point. It could be possible but also maybe things were normal until this happened. Not really sure what they relationship was like before then.
As for the show, I had to look it up because I've never heard of it before. I just kind of skimmed through some parts so I'm not really familiar with the show but I'm not entirely sure what the point of the show is. It seems like it's trying to be optimistic according to it's tone, I guess? Either way, I guess it depends on how he personally feels about the show.
But I'm not sure what's supposed to be funny by the end. It just seems like the joke is "He can't get a date, so let's compare him to the people on a show with disabilities." If he has a disability, that's probably only going to make him feel worse about it. If he doesn't, then what they're saying is that he's so bad at dating that he might as well have a disability because there's something wrong with him.
I think if I knew more about the context of their relationship and what kind of jokes they usually do anyway, maybe I can understand it more, but I think the matter is that a lot of people don't like to be made fun of for certain things and that's really about it.