Smart ridesharing system (SRS) is a ridesharing system integrated with information and communication technology. We formulate a passenger matching problem by considering user preference in SRS. This study aims to investigate day-to-day dynamics and characteristics of passenger matching problem in SRS. A modified stable roommates problem is adopted to represent this matching problem. The characteristics of day-to-day dynamics (e.g., existence, uniqueness, and stability of equilibria) of SRS are understood and revealed by a simulation of formulated model. Through the simulations, the existence of multiple day-to-day equilibria is investigated under some conditions. One of the existing equilibria is an undesirable equilibrium where SRS is not feasible to be operated as no one rideshares, while the existence of a desirable stable equilibrium, where SRS is feasible to be operated, is realized under some conditions.