The condition of parameter identifiability is essential for the consistency of all estimators and is often challenging to prove. As a consequence, this condition is often assumed for simplicity although this may not be straightforward to assume for a variety of model settings. In this paper we deal with a particular class of models that we refer to as "latent" models which can be defined as models made by the sum of underlying models, such as a variety of linear state-space models for time series. These models are of great importance in many fields, from ecology to engineering, and in this paper we prove the identifiability of a wide class of (second-order stationary) latent time series and spatial models and discuss what this implies for some extremum estimators, thereby reducing the conditions for their consistency to some very basic regularity conditions. Finally, a specific focus is given to the Generalized Method of Wavelet Moments estimator which is also able to estimate intrinsically second-order stationary models.