This paper concerns with the identifiability of a vector of unknown deterministic parameters. In many practical applications, the data model is affected by additional random parameters whose estimation is not strictly required, the so-called nuisance parameters. In these cases, the classical definition of identifiability, which requires the calculation of the Fisher Information Matrix (FIM) and of its rank, is often difficult or impossible to perform. Instead, the Modified Fisher Information Matrix (MFIM) can be computed. We generalize the main results on the identifiability problem to take the presence of random nuisance parameters into account. We provide an alternative definition of identifiability that can be always applied but that is weaker than the classical definition, and we investigate the relationships between the identifiability condition and the MFIM. Finally, we apply the obtained results to the identifiability in presence of nuisance parameters to the relative grid-locking problem for netted radar system.