Output-only methods for modal identification are only strictly valid if a set of requirements are fulfilled regarding both structural and environmental conditions. A particularly challenging effect in wind turbine dynamics is the significant presence of nonlinear damping sources coming from aerodynamic forces and, in offshore applications, hydrodynamic forces on the substructure. In this work, the impact of these terms is firstly discussed in analytical terms, and then the corresponding effect on the performance of the covariance-driven stochastic subspace identification is evaluated on a single-degree-of-freedom model. The analysis is then extended to a full hydro-aeroelastic simulation of a 5 MW floating wind turbine using the open source software OpenFAST, mimicking the structural response in free decay tests and in parked conditions with turbulent wind fields. The results show that output-only identification methods are applicable in these challenging scenarios, but the results obtained must be carefully interpreted, since their dependence on the environmental conditions and motion amplitude imply that they are not directly translated into the structure properties, although still closely related to them.