One can't give a complete answer to this question without first understanding *how* etale cohomology does give a cohomological interpretation of the zeta function in the function field case.

Let $X$ be a smooth projective curve over a finite field $\mathbb F_q$. Then

$$ \zeta_{X}(s) = \frac{ \det (1 - \operatorname{Frob}_q q^{-s}, H^1(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell))}{\det (1 - \operatorname{Frob}_q q^{-s}, H^0(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell)) \cdot \det (1 - \operatorname{Frob}_q q^{-s}, H^2(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell))}$$

It is crucial that, to get this formula, one passes to the curve $X_{\overline{\mathbb F_q}}$ obtained by base-changing to the algebraic closure. The reason this is important to note is that there is no analogue of "the algebraic closure of the base finite field" after replacing $X$ with $\operatorname{Spec} \mathbb Z$, at least not in the usual world of schemes.

An optimistic guess is that it suffices to take the etale cohomology of $\operatorname{Spec} \mathbb Z$. If that were true, then it would presumably likewise suffice in the function field world to take the etale cohomology of $X$, without base-changing to the algebraic closure. What happens when we do this?

We can see this using the long exact sequence

$$ \to H^i(X, \mathbb Z_\ell) \to H^i (X_{\overline{\mathbb F_q}}, \mathbb Z_\ell) \to H^i (X_{\overline{\mathbb F_q}}, \mathbb Z_\ell) \to$$

where the third arrow denotes the map $\operatorname{Frob}_q-1$. Examining this exact sequence and the known description of the action of Frobenius on $H^0$ and $H^2$, we see that

$$H^0 (X, \mathbb Z_\ell) = \mathbb Z_\ell$$
$$ H^1 (X, \mathbb Z_\ell)= \mathbb Z_\ell$$
$$H^2(X, \mathbb Z_\ell) = H^1(X_{\overline{\mathbb F_q}}, \mathbb Z_\ell)/ (1 - \operatorname{Frob}_q $$
$$H^3(X, \mathbb Z_\ell) = \mathbb Z_\ell/ (q-1)$$

The only interesting cohomology group here is $H_2$, which is a finite abelian $\ell$-group. Its order is the maximum power of $\ell$ dividing the determinant of $\operatorname{Frob}_q-1$ acting on $H^1$. Equivalently, this is the maximum power of $\ell$ dividing the residue of the zeta function of $X$ at $s=1$.

Combining this information for all primes $\ell$, we have a cohomological interpretation of the residue of the zeta function at $s=1$ (at least up to a power of $p$).

Similarly, over rings of integesr of number fields, not necessarily $\mathbb Z$, we can obtain a cohomological intepretation of the residue of the zeta function. This cohomology group turns out to be dual to the $\ell$-part of the class group, so this is just the Dirichlet class number formula.

There is one last thing you can do here. The field of functions on $X_{\overline{\mathbb F_q}}$ is a Galois extension of the field of functions on $X$, with Galois group $\prod_p \mathbb Z_p$. We can just pick an extension of $\mathbb Q$ with Galois group $\prod_p \mathbb Z_p$, or even just $\mathbb Z_p$, pretend that this is $X_{\overline{\mathbb F_q}}$, and take etale cohomology of its ring of integers.

This, modulo technical details, is the approach of *Iwasawa theory*, which gives a cohomological interpretation of *$p$-adic $L$-functions*. Some $p$-adic $L$-functions are related to special values of the Riemann zeta function at negative integers, so there is, in a sense, a cohomological interpretation of these specific values.

You will note some synchrony here with Peter Scholze's point in the comments that etale cohomology is only suitable for constructing $\ell$-adic $L$-functions, and is only able to construct complex-analytic ones in the function field setting by a trick using the fact that the $L$-functions are polynomials.