Palmolive bath soap in my grandmother's day was a true lye soap made with palm and olive, but the
modern Palmolive dish soap is a detergent blend. I don't think Palmolive bath soap is even on the market anymore (at least in the US).
Could you make a dish soap with palm and olive? Sure. Catch is I don't know how well it would deal with greasy dishes -- you might have to use quite a bit of the soap, more elbow grease, and warmer water to get good results. Coconut oil is usually found in soap recipes for dish, laundry, and household cleaning because CO soap is good at grease cutting.
I use Irish Lass' recipe of CO, OO, and castor at the kitchen sink for washing hands and for light dish washing. This recipe has 25% coconut oil, but it's not nearly as good as Dawn detergent for cutting heavy grease. That said, the homemade stuff works fine on the usual kind of dirty dishes -- that might be 80% of the dishes I hand wash. I keep Dawn on hand for those really greasy and dirty pots and pans. It's amazing how long a bottle of Dawn lasts me nowadays!