Whenever Star Trek characters wear shiny clothes, they read as polyester to us, but I think they’re more like really fancy bedsheets.
Cotton, linen, and even wool can be shiny. Shine is related to how long and thin the underlying fibers are. Any time there’s thicker fibers or a break in the fibers, it casts shadows and scatters light, creating a more matte finish. Longer, thinner fibers lie flatter, bouncing light straight back to the source, creating a more reflective surface.
The problem, of course, is that long, thin fibers are difficult to handle without breaking them. Even after you sort out all the staples that got cut short during shearing/harvesting, the long staples have to survive the rest of the processing and weaving. This is why we assume shiny fabric must be synthetic (or silk). These fibers begin as long continuous strands and are strong enough to stay unbroken as machines move them around.
We can make shiny cotton, it’s just difficult and expensive. And the shine draws attention to any stains, runs, fuzzing, and other tiny defects, so even though longer-staple fabrics are technically stronger, they start to look shabby if they aren’t maintained very carefully. So we usually save this difficult and expensive process for bedsheets and formalwear, where we can benefit from the increased quality without having to maintain its flawless appearance over daily wear.
Both of these problems are solved by replicators. A replicator isn’t manipulating very long, very fine staples and trying not to break them, it can just materialize them into their final positions. And if those fine fibers get damaged or stained, just pop that corner into a replicator and it’s good as new.
In a world where it’s much easier to make (and maintain) high-quality fabric, shiny clothes wouldn’t look as “costume-y” to other people. They’d see someone in shiny fabric and assume they just wanted to wear something soft and comfortable, the way we’d look at basic t-shirt fabric today.