Someone in my Facebook Enneagram Type 4 group asked how other Fours dealt with people who are constantly reminding them of how much they were similar. She said she had a friend who was always pointing out their similarities. My first guess is because this poster is a Four, that the other person is too. Generally people who are the same personality type aren't super attracted to each other- they'll be polite, but there's nothing to project onto, so on an egoic level, they're just taking up each others' space. Especially because they're both image types, they're fighting for the same response from others- and from each other, which makes conversation a bit stilted.
The question reminded me of when I went to my first Enneagram workshop at the Enneagram Institute, and I met a Type 8 who said that by the middle of the week, all the 8's were evenly spaced out along the room- they were all defending their territory; none of them wanted to get close to each other. I mean, this is a generality- the same type can be attracted to each other sometimes, and even marry. At my second Enneagram workshop, I met two married couples who were composed to two Fours each, and that is a rare combination among married couples of the same type. There aren't many 4-4 couples.
The other night I went out for drinks with another Four and I tried to be aware of my tendency to point out our similarities. He didn't know the Enneagram and I was trying to just let him be, trying to be non-reactive to my craving to jump in and do a Type Four intervention on his life. Fours especially need to feel unique, and nobody likes being put in a box- they want to feel like the other person is being present to their whole self, not just the characteristics that are consistent with the archetype. I took it as an opportunity to see what triggered me. We don't like being reminded of our shortcomings, and when we see them in another one of our kind, we want to push them away or "help them change". I think the only way to learn to get along with someone with the exact same personality as you is to learn to love yourself.