Robustness of stability with respect to external perturbations is an important property characterising the ability of the dynamics to counteract the influence of uncertainties. In the present paper, such a property is investigated for the class of passive and strictly passive systems, which have several invariant compact and globally attracting subsets in the unforced scenario. It is assumed that the storage and supply rate functions are sign-definite with respect to these sets. The results are obtained within the framework of input-to-state stability and integral input-to-state stability for multistable systems. The robustness conditions are obtained for open-loop and for closed-loop cases, i.e. when an output feedback is required to guarantee robustness. Two applications (related to the model of multispecies populations) of the proposed theory are used to illustrate its efficiency.