As far as I know, it is impossible to use laundry soap for making grease. I just wanted to try to see what happened. Anyway, most greases are an oil, mineral oil, synthetic oil, or an "edible" oil, (soy bean oil, canola oil, or whatever) mixed with a soap. And the soaps are most often a lithium soap, which is also called lithium stearate. Then it seems to be calcium soaps that is a good number two, and it is also possible to use aluminum and/or magnesium soaps. All these mentioned soaps comes as a white powder. The soap and oil is heated, and the oil will thicken. It is of course a lot more to it than that, what type of oil to use, how much oil, and how much soap for example. The end result can be a fairly thin grease, or a thicker grease. These "soaps" I have mentioned is also called stearate, like lithium stearate, calcium stearate and so forth. The soaps are not washing soaps in the sense we think about for washing our hands and so forth.
For example calcium stearate is not dangerous, it is even used as an additive in food. And mixed with an edible oil you will get food grade grease. Mixed with mineral oil, like a gear oil, it will not be a food grade oil anymore. So, as far as I have read and found out, all greases are based on an oil. Just mixed with something that makes it thicker.
Some greases are made of oil thickened with something else than soaps, for example is aircraft greases often thickened with a clay. And it is even possible to mix an oil with a petroleum jelly to make it thicker. And, a couple of places I have seen it mentioned to use washing soda. But, I don't know what kind of washing soda.