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.)

Is It Possible To Make Liquid Soap Out Of The Left Overs Of A Heap Of Toilet Soaps?

is it possible to make liquid soap out of the left overs of a heap of toilet soaps ,adding glycerine and the like?