OK, then I think we’re on the same page. The likelihood isn’t classically identified, but with a prior, you get identification. The prior gives you identification even without data. Andrew has several notions of identification just in his book with Jennifer—you need to generalize to Bayes and to practice where you actually care about particular data sets.
To be clearer about what Stan’s trying to do, it unconstrains the parameters to have support on R^N. It assumes that it has support over all that, not just over a lower-dimensional submanifold. Stna will let you build the above model
a ~ foo(...);
b ~ bar(...);
x = a + b;
x ~ baz(...);
The real problem is that in this simple case, a and b are only identified through their priors.
Stan tries to unconstrain the parameters to have support on R^N. We don’t allow hard relations that produce submanifolds as there’s no way to produce a volume-preserving Jacobian. So we could parameterize this in terms of a and b as above and x could be defined as a transformed parameter.