Just to add to the confusion: Monoidal categories are the appropriate notion of "monoid object" in the category of categories; unit and associativity law are re

Just to add to the confusion: Monoidal categories are the appropriate notion of "monoid object" in the category of categories; unit and associativity law are replaced by coherent natural isomorphisms (see "categorification"). So that's where the "monoidal" comes from.