> We can also talk about a monoid object in a larger category in much the same way.
Indeed that is one way of viewing a monoid. However, the most basic categorical definition of a monoid (and the first one usually introduced in a category theory textbook) views it as a category with only one object, where all the "interesting stuff" happens with the morphisms. A monoidal category and monoid objects are a restatement of that idea in a larger category.
