The Tarahumara are the fastest long-distance runners in the world. They don’t run completely barefoot, but that’s practically the case (they wear a kind of sandal made out of old tire). Sponsors have tried getting them entered in professional races, but it did not go so well; the Tarahumara runner took off his sponsor’s shoes early in the race, chucked 'em, and ran the rest of the race barefoot (beating the nearest competitor by a wide margin).
Because of that, many professional/conventional trainers now have their running athletes train barefoot. The result is actually fewer injuries, not more. It conditions the foot to be stronger and more resistant to injuries.
So when you practice fighting games, take off your shoes.
And don’t worry so much about the slight slipping and sliding of the stick on your surface. It might actually make your game better, as you condition yourself not to over-commit too strongly to any one direction/button press. Timing and deciding to press the button/direction you want is the important thing, not how hard you do it. Instead of relying too much on the weight of the stick, focus more on being able to quickly find your neutral or diagonals, WITHOUT pressing too much so it would move the entire joystick itself.
You know how like many people esp. non-gamers hold a console’s joypad and move the joypad itself in the air while playing, as if that would have some effect on the game’s character? like when they try a big long jump they kind of hold their joypad up a little higher in the air or more in the direction they want to jump to. It’s the same kind of thing and you want to reduce it as much as possible.