The smell in a climbing or bouldering gym is because many climbers (and most climbers above a beginner/intermediate level) are probably taking OFF their shoes when not actively climbing. You do this outside, too -- the shoes are TIGHT and pretty uncomfortable to stand around in, so you only wear them when you're on the rock.

Climbing and bouldering are pretty intense, though, so you will get sweaty. And you'll sweat in your shoes. And the shoes will get stinky. Shoe stink is often somewhat contained if your foot is still IN the shoe, but if you take them off everyone gets to enjoy the aroma.

Unlined leather shoes handle the funk the best. OTOH, shoes with uppers made of textiles, especially when lined, end up being de facto bioweapons. I am not cursed with especially stinky sweat or feet, but I had a pair of fabric-lined climbing shoes that had to ride in the trunk going to and from the gym or the crag because having them inside the passenger compartment of the car was absolutely untenable.

Yoga, OTOH, is done barefoot. People often show up very minimal shoes. There's a sweat smell in many yoga spaces, especially hot yoga spaces, but it's not the funk associated with shoes.