What does it mean to identify structural shocks in macro models?

A great up to date treatment that I keep coming back to is Ramey (2016). It discusses identification in both articles you refer to. The short answer to your question is that obtaining structural shocks is always based on theoretical reasoning and assumption imposition including in both of your examples (e.g. Christiano Eichenbaum Evans (1999), Gali (1999)).

On page 5 Ramey writes:

I view the shocks we seek to estimate as the empirical counterparts to the shocks we discuss in our theories, such as shocks to technology, monetary policy, fiscal policy, etc. Therefore, the shocks should have the following characteristics:

(1) they should be exogenous with respect to the other current and lagged endogenous variables in the model;

(2) they should be uncorrelated with other exogenous shocks; otherwise, we cannot identify the unique causal effects of one exogenous shock relative to another; and

(3) they should represent either unanticipated movements in exogenous variables or news about future movements in exogenous variables.

Any shock satisfying (1) - (3) can be used to make statements about the causal effect of an unexpected change to a variable on contemporary and future values of all variables in the model, which is the main goal of contemporary econometrics.

It seems that in earlier times researchers were more cavalier with identification, which is reflected in for example the oirf procedure in STATA, which will happily estimate "causal" effects based on orthogonalizing the residuals from the reduced form model, but this reasoning appears to be outdated. I think its contemporary version are sign restrictions a la Uhlig (2005), which also stem from theory.