I've always associated 'padding around' with (relatively) noiseless steps and assumed it came from 'pad' for animal paws.
So theoretically it would be possible to pad around in ballet slippers or the equivalent, which I guess is why Bel thinks (like me) that you can wear socks and still pad.

...irrelevant thought from association with sneaking up on prey / padding around silently like a thief / socks vs bare feet: - are toeprints as unique as fingerprints? I would assume so. Have they ever been used in crime suspect identification?