So according to Mitsch, the natural partial order $\leq$ of any semigroup $S$ is given by $$a \leq b \iff a = xb = by, xa = a, \quad \text{for some } x, y \in S^1$$ But obviously in $(\mathbb N, \cdot)$ we have $2 \leq 3$, and yet there is no $x \in \mathbb N$ that satisfies the above condition.
Similarly in $(\mathbb N, +)$ we have $2 \leq 3$ and the same problem.
Is there something that I'm missing? Is regular "less than or equal" not a natural partial order on $\mathbb N$?
Mitsch is defining one particular partial order on any semigroup and calling it "the natural partial order". This is nothing more than a name; it is not an assertion that any other partial orders are "unnnatural" in any sense. As you have observed, in the case of $(\mathbb{N},\cdot)$ or $(\mathbb{N},+)$, it is not the same as the usual partial order of natural numbers.