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Linearized fluid equations for the collective modes of weakly inhomogeneous plasmas including strong coupling effects are derived from a recent kinetic theory based on an extended Singwi-Tosi-Land-Sjölander (STLS) ansatz [H. Kählert, G. J. Kalman, and M. Bonitz, Phys. Rev. E 90, 011101(R) (2014)]. The equations are analogous to the equations of linearized elasticity theory with space dependent elastic moduli that correspond to those of a bulk system with the local fluid density. The identification of the latter as infinite-frequency elastic moduli shows that the present version of the extended STLS theory accounts for the elastic properties of the fluid but does not capture the viscous behavior. [ABSTRACT FROM AUTHOR]