I have made the same observation as you Jaaret. I feel that the cheap store soap can leave your hands dry and stripped, but I do believe too that handmade soap (superfatted with oils) can leave an oil residue which can leave skin feeling less dry and stripped. Although soap helps remove soils and bacteria but trapping them in micelles (
http://en.wikipedia.org/wiki/Micelle) and then they are physically removed by running water, I don't think it is a total removal process, as our hands are not sterile after using soap, just a bit less gunky.
I think it is something that has to be thought of on a microscopic scale. The surface of the hand is rough, with crags, crevasses, and hairs as well lipid based cell membranes which would be more likely than water (the water you are washing with) to attract/hold oil molecules. I think additives to soaps can also be beneficial because they are infused into the oils which remain on the skin after washing.
Goats milk soap is an example of soap that has done wonderful things for people's skin, and I believe it is because some of the chemicals from the goats milk are left behind (to be absorbed) in the supper fatted oils.
For example of you were to dunk your hands in olive oil (I do this when cooking sometimes), and then go wash them with soap, it is very hard to get the oil off as it is much more attracted to your skin. I can usually never get it all off, but it makes my skin feel very nice and eventually soaks in and rubs off.
Similarly with washing oil off of plastic, as plastic is oil based and oils are more attracted at the plastic and very hard to wash off, even with detergent.
Just my two cents
I think things look more straight forward on a macro level (soap=clean), but on a microscopic level are more complicated.
Have a good week!