What Name Is Animals Fat Used To Make Soap And Washing-up Liquid Hidden Behind?

Do they need to say they use animal fat to make soap? Or is it a case if it doesn’t say it doesn’t use animal fat, it definitely has animal fat in the ingredient?
How about Fairy Liquid? (Sorry America, it’s a British washing-up liquid. I am not trying to be flipp.)

What Is This Easiest, Quickest Way To Make Scented Soap And Shampoo?

I’ve looked for soap and shampoo making recipes, but they all seem so difficult. I don’t want to make soap from scratch; I just want to find an unscented soap I can add essential oils to. Is this possible?