There would be an isomorphism if you considered $\mathbb Z \cup \infty$ instead of $\mathbb N \cup \infty$. Perhaps you meant to ask about it. Perhaps you also wanted to ask a question about $(\mathbb N \cup \infty, \min, +)$, because it's the semiring in the definition of discrete valuation, and maybe about $(\mathbb N \cup -\infty, \max, +)$ for symmetry.

It isn't true that $(\mathbb N \cup \infty,\max,+)$ and $(\mathbb N \cup -\infty,\min,+)$ are isomorphic: both have infinity as the absorbing element w.r.t. $(\max, +)$ and $(\min,+)$ respectively, so an isomorphism would map $\infty \mapsto -\infty$ and $\mathbb N \mapsto \mathbb N$. However, $0$ is an identity element for $\max$ in the former ring, while the latter doesn't have an identity element for $\min$.