> So how one achieves this magic number 7?

The fact that monoids have to be associative heavily constrains the set of possibilities. [This blog post](https://kavigupta.org/2016/07/20/Counting-Monoids/) goes into some detail about counting monoids and similar structures. They give a formula for "pseudomonoids" (which are really [pointed magmas](https://en.wikipedia.org/wiki/Magma_(algebra))) as \$$n \cdot n^{n \cdot n}\$$ (read: \$$n\$$ choices of point and \$$n^{n \cdot n}\$$ choices of binary operator), which can be adapted to account for the behavior of the (fixed) unit as \$$n^{(n-1) \cdot (n-1)}\$$. When \$$n = 3\$$, this gives exactly your result of \$$81\$$, showing that the objects you're counting need not be associative!