My experience with indoor cycling is that the majority of the people have never been on a road bike, wear gym styled athletic clothes, and think outdoor cyclists are weird! It was not like this when I first went to spin classes 10 years ago. Soul Cycle seems a little cultish, so I could understand if you feel different!
I wear my regular cycling shorts, but I also have a pair (and some capris) with a very thin chamois that i use for commuting and look less like road wear. I do wear jerseys, but I have found wearing a sleeveless one makes you stand out less and are just cooler for the steamy atmosphere. You could always wear a regular athletic tank for spin. I get those from Athleta.