Because pockets were worn by people in every layer of society historically what they were made out of varied intensely. Some were made out of sturdy and plain fabrics, marked with the owners initials. Others were made out of lavish silks or richly embroidered.
As such when it comes to fabric options the world is your oyster.
The one thing to keep in mind is how stable your fabric is. If it is not very stable or you think it won’t hold very much weight, you can add a lining.
Your binding doesn’t have to be the same colour as the main fabric either. You can use it to add a fun accent, or make your pocket fancier. Bias binding curves easier around any curves, but you can just as easily use strips of fabric as a more waste conscious option.