I’ve been out of the business for a while now, but “back in the day” when they first came out, a hand soap would be much milder in terms of detergent component and also normally has more of a fragrance added. Dish soaps have stronger surfactants (detergents) and usually the formula is designed to remove oils. Oil on your skin, even your hands, is what keeps them soft and not dry. So if you don’t mind having dry, cracked skin on your hands, you can use dish soap for your hands. I personally wouldn’t use hand soap for my dishes as it might make them smell strongly of perfume (depending on the soap) which would make your food taste “off”, plus you might need a lot more to get your dishes clean.
I’d also suggest that the use of anti-bacterial anything at home is of debatable merit.