Edit (following the clarifications of the OP): This is not an answer, it is rather a long comment.
Studying this kind of questions is part of the mission of factorization theory: The language of the theory (at least in its classical incarnation) is the language of commutative (and, to a very large extent, cancellative) monoids, which seems to fit with the OP's idea of a monoid embedding $f$ from the multiplicative monoid of the positive integers into a larger commutative monoid $S$ such that $f(p)$ is a product of atoms${}^{\text{(a)}}$ in an essentially unique way. Note that, in fact, we may assume without loss of generality that $S$ is an atomic monoid (i.e., every non-unit of $S$ factors as a product of atoms).
Now, it is a basic result in the classical theory of factorization that a commutative cancellative monoid $H$ is a unique factorization monoid (i.e., every non-unit of $H$ has an essentially unique factorization into atoms), if and only if $H$ is atomic and every atom is a prime${}^{\text{(b)}}$, if and only the quotient monoid $H/H^\times$ is a free abelian monoid; for a reference, see Theorem 1.2.9 in
A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, 2006).
This means that, if $S$ is a cancellative and commutative monoid with the unique factorization property, then there is nothing new we can hope to learn about the arithmetic (multiplicative) structure of $\mathbb N$ from the (mere) existence of the embedding $f$ — and this would be true independently of how condition (3) is interpreted. So, for the question to make sense, we need either to give up the idea that $S$ is a cancellative and unique factorization monoid (which seems to be fine with the OP), or to keep track of the additive structure of $\mathbb N$ and rather look, say, for a semiring embedding of $\mathbb N$ into a larger commutative semiring whose multiplicative monoid satisfies condition (2).
• Notes.
(a) An atom of a monoid $H$ is a non-unit $a \in H$ such that $a \ne xy$ for all non-units $x, y \in H$.
(b) A prime of a monoid $H$ is a non-unit $p \in H$ such that, if $p \mid_H xy$ for some $x, y \in H$, then $p \mid_H x$ or $p \mid_H y$ (here $\mid_H$ is the divisibility preorder on $H$, so $a \mid_H b$, for some $a, b \in H$, if and only if $b \in HaH$).