The definition of a monoid here is not the usual definition, is a new definition for the special case of a strict category as defined MacLane's Book Categories for the Working Mathematicean. Since you maybe thinking about composition of endofunctors and unit endofunctors you get a confused picture. The monad as a monoid in the category of endofunctors is a way to show that you can confuse people using two different definition of an usual concept and both give different results. I got this from (1), look for "main confusion": monoid in the category of endofunctors is defined in a new way, and is not the expected monoid in the set of all endofunctors.
The definition of monoid in monomial categories is on page 166 and Monads in a category are on page 133. As a math person I know what is a monoid (usual term) but I did not know what is a monoid in a monomial category (well I know now because is on page 166 of the book).
(1) https://stackoverflow.com/questions/3870088/a-monad-is-just-...