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?

Yes, it needs to hold for all $z$ (subject to some technical caveats)
The idea of non-identifiability is that two different parameter values give the same sampling distribution, making them impossible to distinguish based on the data.  Subject to some technical caveats which you can usually ignore,$^\dagger$ this means that non-identifiability occurs when the sampling density is the same for all observable data values and identifiability occurs when the sampling density is different for at least one observable data value.
Consequently, in the definition you cite in your question, I think they mean to say that identifiability occurs when $f(z|\theta) \neq f(z|\theta_0)$ for some observable value $z$.  Your attempted counter-example gives one case where $f(z|\theta) = f(z|\theta_0)$ but it does not show that this holds for all $z$, so it does not establish non-identifiability.
Incidentally, a reasonable way to view identifiability is in terms of the concept of minimal sufficient parameters (see e.g., O'Neill 2005).  Just as you can derive a minimal sufficient statistic from the likelihood function, you can similarly derive a "minimal sufficient parameter" by the same essential method.  The minimal sufficient parameter is what can be "identified" from data from that sampling distribution, so any parameter vector that is not a function of the minimal sufficient parameter is not fully identifiable.

$^\dagger$ A slight complication to identifiability occurs because density functions are not generally unique representations of probability distributions.  For instance, for continuous random variables it is possible to alter the points in a density function on an arbitrary countable set of points and it still represents the same distribution.  This means that when you are assessing identifiability based on a parameterised class of sampling density functions, strictly speaking, identifiability occurs when $f(x|\theta) \neq f(x|\theta_0)$ over a set of values of $z$ that has positive probability under at least one of those densities.  If you form the sampling densities so that they are all continuous then this is enough to allow you to simplify things to say that identifiability occurs when $f(x|\theta) \neq f(x|\theta_0)$ for any $z$.  For the reasons discussed here, identifiability is generally not defined in terms of density functions.
