In this work, we study the extended viscous dark energy models in the context of matter perturbations. To do this, we assume an alternative interpretation of the flat Friedmann-Lema^itre-Robertson-Walker Universe, through the nonadditive entropy and the viscous dark energy. We implement the relativistic equations to obtain the growth of matter fluctuations for a smooth version of dark energy. As result, we show that the matter density contrast evolves similarly to the $Lambda$CDM model in high redshift; in late time, it is slightly different from the standard model. Using the latest geometrical and growth rate observational data, we carry out a Bayesian analysis to constrain parameters and compare models. We see that our viscous models are compatible with cosmological probes, and the $Lambda$CDM recovered with a $1sigma$ confidence level. The viscous dark energy models relieve the tension of $H_0$ in $2 sim 3 sigma$. Yet, by involving the $sigma_8$ tension, some models can alleviate it. In the model selection framework, the data discards the extended viscous dark energy models.
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