Is the FSI analysis always considered Nonlinear or the solid part analysis can be linear too?
IMHO this is a matter of convention more than anything else. However, I acknowledge that there are other more experienced and knowledgeable people in this forum (e.g. alephzero), that can offer a more scientific (and less based on experience) view on the subject.
My view is that Even if the solid part's constitutive equations are elastic, the interaction between fluid and solid is always time dependent, and has sinusoidal components of response even in the steady state. Therefore, I would consider it as non-linear dynamic.
However, there maybe some cases, where if you don't have any vibrations (not many for real FSI problems). In that case, some (I'm not one of them ) may chose to consider the problem as linear.