I think it's also a low-IQ version of Stephen Potter's "two-club man" gambit, which I think first appeared in his satirical self-help work, Gamesmanship.
Let's say you're a mid-century British gentleman. You decide to join two clubs, one for military men, one for artists. When you attend the military club, you dress and act like an artist. When you're at the artist's club, you assume a military poise. In each case, you get to stand out. You can't excel in competitive company, so you can seem extraordinary as the odd man out. The artists think you must be quite the soldier; the military men think you're a serious artist.
Patrick may hope that he can impress people who know jack shit about military matters by palling around with people who don't. Maybe he hopes to find more of those people on the left than the right. He's quite wrong, of course; military veterans have many political leanings. But would it be Rick if he didn't get it wrong?