The competition in the ride-hailing market can influence traffic congestion and deteriorate the quality of service. A customer can request to be matched with a ride-hailing vehicle from their preferred company, which might not be the nearest vacant vehicle. This can increase both the customer’s matching and pick-up waiting time and the vehicle’s travel distance to the customer, and contribute to traffic congestion. Recent studies focus on the long-term competition effect by considering network equilibrium. In this work, we target a shorter timeframe and investigate how competition influences the passenger–driver matching process, the consequent vehicle travel to the customer, and more globally the system at the operational level. To this end, we propose a modeling and simulation framework based on the Macroscopic Fundamental Diagram (MFD). We apply the so-called M-model, a continuum approximation of the trip-based MFD. Compared with the accumulation-based approach, it explicitly monitors the remaining travel distance of all vehicles. We extend the mathematical M-model decomposition and focus on accurate dynamic estimation of trip lengths for the different vehicle states based on the immediate system state. For this, we suggest creating an additional proxy simulation framework replicating the demand requests and the service vehicle movements. We propose calibrating the matching function by sampling observations on a proxy grid network. Finally, we assess and compare different matching processes that define diverse competition scenarios: competition, cooperation, and competition with partial cooperation (coopetition). The cooperation scenario shows the best results in service performance.