Identifiability of Non-Parametric Utility Function?

I recently learned that EU characterized by independence and weak ordering is identifiable, but a utility function like: $$U(x)=v_1(x)v_2(x)$$ is not identifiable.

Does it mean that "cardinal uniqueness = identifiability"? Or cardinal uniqueness implies identifiability but not the other way?

I found the precise definition of "identifiability" parametric function. But for non-parameteric function, what is the precise definition of identifiability?

For example, does the simplest utility function $$u(x)$$ characterized by weak order and continuity identifiable by its own? (a.k.a does ordinal uniqueness implies identifiability?)

• So what is the definition of identifiability for parametric functions and in what sense is EU parametric? Dec 1 '20 at 16:09
• @MichaelGreinecker EU is non-parametric in this case. Dec 1 '20 at 16:34
• That's not what I asked. Dec 1 '20 at 17:15
• @MichaelGreinecker For parametric functions, identifiable means that any two different parameter vectors cannot give the same value. Dec 1 '20 at 17:17
• Non-parametric does not mean there are no parameters, it usually means the parameter space is infinite-dimensional (which would also be the case with EU with infinitely many outcomes). So the same definition works in a non-parametric setting. Dec 1 '20 at 17:30