It is a bit silly for a joint as snobby as Radiant Raiment to be selling pre-blemished clothing, but the dirt on everyone else's threads is to be expected considering the time period. Nirn's pretty much medieval, and in medieval times laundry was an entirely different affair than it is today. Soap was a rarity, for one. To get the filth out of the fabric, they'd dunk it in a river and beat it with a washing bat or scrub it over a ridged board. It definitely wasn't a weekly thing, either. Laundry took up precious time that was better devoted to chores, so they'd only have washing sessions every few weeks or even months. Even then, outer garments - the ones you see Skyrim's citizens wearing - weren't cleaned all that often. Usually they'd just hose down the inner garments, underwear and the like. Coloured clothing was washed even more rarely than plain linens since the harsh treatment faded the dye, and seeing as most of the outfits worn by the NPCs are dyed... you see what I'm getting at. The dirtiness is perfectly understandable.
And most people would only own 1 or 2 suits of clothing which they or a member of their family made and would be patched, mended, handed down etc until they fell apart
but thats most people. Those who could afford to buy clothes from a shop would have servants to wash/mend them or could afford to send them out to be laundered. Dirty fine clothes is a contradiction in terms. Once they are beyond washing the jarl or thane or merchant would pass them on to a servant.