In game it only uses one texture for all the leather bits, so in game the belt does have a slight texture as well. In some of the hi-res renders/promo images though the belts appear smooth, so you could argue that the belt should be smooth, and they just didn't bother giving it a smooth texture in-game since it was a minor detail. IRL you don't usually see textured leather belts because that kind of leather tends to be soft and and stretchy.
The in game model does include the rivet - no other holes, but I think that's mainly because its a magical belt loop with no seams or adjustments.