We introduce an improved algorithm for the dynamic taxi sharing problem, i.e. a dispatcher that schedules a fleet of shared taxis as it is used by services like UberXShare and Lyft Shared. We speed up the basic online algorithm that looks for all possible insertions of a new customer into a set of existing routes, we generalize the objective function, and we efficiently support a large number of possible pick-up and drop-off locations. This lays an algorithmic foundation for taxi sharing systems with higher vehicle occupancy - enabling greatly reduced cost and ecological impact at comparable service quality. We find that our algorithm computes assignments between vehicles and riders several times faster than a previous state-of-the-art approach. Further, we observe that allowing meeting points for vehicles and riders can reduce the operating cost of vehicle fleets by up to 15% while also reducing rider wait and trip times.