Formal Definition of Identification 
This definition of identification (the bracketed part) is confusing to me because (based on my obvious misunderstanding) it fails for probit:
Probit with 2 covariates: $f=\Theta(X_1\theta_1+X_2\theta_2)^y\cdot(1-\Theta(X_1\theta_1+X_2\theta_2))^{1-y}$. If we fix the data (which might be the mistake) such that $y=1,X_1=1,X_2=1$ and suppose $\theta_0=(1,1)$ then it's easy to find $\theta'=(2,0)$ such that $f(z|\theta_0)=f(z|\theta')$.
What am I not getting here? Does it need to hold for all z in the data generating process?

I see no problem with how identification is defined. The bracketed part is saying that $\theta_0$ is identified if whenever $\theta \neq \theta_0$ and $\theta \in \Theta$, then it must be that $f(z|\theta) \neq f(z|\theta_0)$.
Here, the $\neq$ is for two functions. Two functions are not equal if there is some $z$ for which they are not equal. So I see no ambiguity in the definition.
If you wanted, you could certainly add "if there exists some $z$ such that...", but I presume the author was trying to keep things concise.
Relating all this to your example, identification deals with population distributions, not finite draws of data as in your example (I write more about this here: What is the difference between Consistency and Identification?). So if you are going to "fix" the data, then identification will be about the corresponding finite sample distribution (which is a bit strange, but can be done). It could certainly be the case that for certain finite sample DGPs, something like the coefficients for a probit are actually not identified.
