What is the correct analogue of $\mathbb N$ in a ring of integers?

My conviction is also that the correct analogue of $\mathbb{N}$ for an arbitrary ring of integers $\mathcal{O}_K$ is the semiring of nonzero ideals. This construction satisfies versions of your desired properties: there is indeed a copy of $\mathbb{N}$ (with the usual multiplication but a different addition), given by the principal ideals $(n), n \in \mathbb{N}$, and the construction also ignores units. One way in which this conviction is borne out in practice is that this is what you sum over to define the Dedekind zeta function, which reduces to a sum over $\mathbb{N}$ when $K = \mathbb{Q}$, where you get the Riemann zeta function.

To my mind, the most important property of $\mathbb{N}$ as a subsemiring of $\mathbb{Z}$ hasn't even been pointed out yet: $\mathbb{Z}$ has a natural order with respect to which $\mathbb{N}$ is the subsemiring of positive elements. This is the order that crucially figures in induction, which is arguably the most important thing we use $\mathbb{N}$ for.

There's no reason for there to be a good analogue of this in an arbitrary number field, since among other things they don't come equipped with natural orderings. (The story of number fields is really about generalizing aspects of $\mathbb{Z}$ and $\mathbb{Q}$, not $\mathbb{N}$.) One way to equip them with orders is to embed $K$ into $\mathbb{R}$, but sometimes (e.g. when $K = \mathbb{Q}(i)$) there are no such embeddings, and sometimes (e.g. when $K = \mathbb{Q}(\sqrt{2})$) there is more than one.

When studying the natural numbers $\mathbb{N}$ in number theory, the key property that they have is not that $\mathbb{N}$ is an "efficient" version of $\mathbb{Z}$ containing exactly one element of each orbit of the action of $\mathbb{Z}^{\times}$ on $\mathbb{N}$ by multiplication. Rather, what is most important about $\mathbb{N}$ in number theory (especially in multiplicative number theory) is that each element of $\mathbb{N}$ factorises into products of powers of primes (i.e. the fundamental theorem of arithmetic).

So as Zhen Lin commented above, the number field analogue of natural numbers are integral ideals, because they satisfy the same factorisation properties. Similarly, integral ideals have a natural ordering given by the absolute norm, which allows one to generalise the notion of summing over the first $n$ natural numbers, by summing over integral ideals of absolute norm at most $n$.

You may be interested to read about Beurling primes, which generalise this even further by thinking of "natural numbers" as the multiplicative semigroup generated by products of powers of an infinite set of "primes", together with an absolute value. It's an active area of study to see what conditions are needed on these generalised primes to still have an analogous form of the prime number theorem.