It seems that
\begin{align} &\prod_{\Omega(n)=2}^{}\dfrac{1}{1 - n^{-s}}\approx\zeta (s)\exp \left(P(s)^2/2-P(s)\right)\\ \end{align}
where $P(s)$ is the prime zeta function, $\Omega(n)$ is the number of prime divisors (with mutiplicity) of $n$, and where the RHS is the dominant term in the expansion of the Euler product.
Is this close enough to be of use in any practical application?
To answer the main question "Is this close enough to be of use in any practical application?" I do not believe so. However, one cannot say "no" with certainty, so it seems unlikely that you will get a concrete answer.
I wanted to add a derivation of a more precise formula of what you gave above. In particular, notice that that $\log\zeta(s)\approx P(s)$, so the $\zeta(s)$ and $\exp(-P(s))$ terms should cancel out in some way. Judging by your previous questions, this may not be new to you, and you likely used this same derivation to arrive at the formula given in your question.
Let $\mathcal{R}=\{{n:\ \Omega(n)=2\}}$ and consider the logarithm of the product above. This equals $$\sum_{n\in \mathcal{R}} -\log(1-n^{-s})=\sum_{n\in \mathcal{R}}\sum_{k=1}^{\infty} \frac{n^{-ks}}{k}.$$ Setting $P_2(s)=\sum_{n\in \mathcal{R}} n^{-s}$, the left hand side above equals then equals $\sum_{k=1}^{\infty} P_2(ks)/k.$ Now, $$P(s)^2+P(2s)=\sum_{p}\sum_{q}(pq)^{-s}+\sum_{p}p^{-2s}$$ $$=2\sum_{p}\sum_{q\geq p} p^{-s}q^{-s},$$ and so $$\sum_{k=1}^{\infty} P_2(ks)/k=\frac{1}{2}\sum_{k=1}^\infty \frac{P(ks)^2+P(2ks)}{k}.$$ Now, $\log\zeta(s)=\sum_{k=1}^\infty P(ks)/k$, so we find that
$$\prod_{n:\Omega(n)=2}\frac{1}{1-n^{-s}}=\sqrt{\zeta(2s)}\exp\left(\frac{1}{2}\sum_{k=1}^\infty P(ks)^2/k\right),$$
Now, in fact, $$\sum_{k=1}^\infty P(ks)^2/k=\sum_{q}\sum_{p}\sum_{k=1}^\infty\frac{q^{-ks}p^{-ks}}{k}$$ $$=\sum_{p}\sum_{q} -\log\left(1-(qp)^{-s}\right),$$ and hence
$$\prod_{n:\Omega(n)=2}\left(1-n^{-s}\right)^{-2}=\zeta(2s)\prod_{p}\prod_{q}\left(1-(pq)^{-s}\right)^{-1}$$ which I will rewrite as
$$\prod_{n:\Omega(n)=2}\left(1-n^{-s}\right)^{-2}=\zeta(2s)\mathcal{P}$$
where $\mathcal{P}$ is precisely the double product appearing in your previous question Zeta function double product.
