I wore jeans and cowboy boots all through college to ride English because I was broke and that's what I had. I was only riding in university classes, and we were in Montana, so while I got looks from the snobby girls from the East coast, the instructor didn't mind.
However, I feel like it is a respect thing. I will wear whatever to ride in my pasture at home, but I try to wear clean full seat breeches, paddock boots, and a tucked in polo at my lesson barn out of respect for the instructor and her beautiful facilities. You wouldn't show up to work in dirty, holey clothes, or something inappropriate for the job for similar reasons.
There's also the issue of damaging an English saddle with the seam on jeans from repeated use. Some newer English saddles have really soft leather, so this can be a problem. Some don't though.
It also really depends on the kind of barn you are at. You could try chatting with your trainer about it. If they actually have somewhat of a dress code, you could probably be compliant with a $20 pair of Dover breeches, a cheap polo from JC Penney, and cheap half chaps over your western boots (depending on what type they are). Some barns are very relaxed and riding in Western attire would not be an issue at all.
Tug on anything at all and you'll find it connected to everything else in the universe.~John Muir