In abstract algebra there really is a hierarchy of group-like-structures with different levels of symmetry (see the table in [1]), and a monoid is one of such structures.
I've had some luck with the first chapter of this (free) [2] book, although they explain groups and not monoids but for me it was then rather quick to gain intuition about monoids.
[1] https://en.wikipedia.org/wiki/Monoid#Relation_to_category_th...