You didn't specify that you want warm weather undershirts, so if you're looking for cool weather - consider the Uniqlo heat tech, but also merino wool. Merino can get expensive, but I have some great tees from Costco a few years back - I think long sleeve were $20 (USD) and short sleeve were $15? The ones I have are a mid-weight, so they're great for cooler weather, but not for the tropics or warm temps.
My husband has a ton of the uniqlo airism and heattech. (as a woman...) I like the heattech but the airism are a little odd on me. And the bra-inclusive versions just plain don't fit
I've not noticed any smell accumulating on the uniqlo (or merino) but it's possible he doesn't have the wear on them that would lead to the smell. I hate to encourage wastefulness/non-sustainable practices, but the uniqlo are cheap enough (esp when on sale, which they are regularly) that they can be replaced if/when they start to smell.