Uniqlo do have Dri-Fit shirts made of the same material as Nike. I bought a few of them and they are same quality as the Addidas ones that I owned previously but only cheaper. Look for them in a separate section.
Just visited Ion & visited the Uniglo store & saw the polo shirts selling for only $16.90. Although I was looking for shirts & not T-shirts I bought a few of them
The material used is something called Supima.
Also visited the Columbia store & saw shirts of the quick dry kind selling for $49 to $80+. They look like normal shirts & you would have to read the labels to find out that they are the quick drying kind.