The 3 points you list are assumptions of the homogeneous Poisson point process. If you are thinking of an inhomogeneous Poisson point process as raised by rohan, where the intensity is allowed to vary, #2 need not be true. But conditioned on the intensity (if known), both #1 and #3 are satisfied by inhomogeneous or homogeneous Poisson point process. In real life data, simultaneity of something like taxi rides is so unlikely, even if it were to occur, it modeling taxi rides (of a fleet of taxis) as point process would still be a good approximation.
Now you're asking whether the accumulated events generated by the point process follow a Poisson count distribution in some interval in time. If the process is homogeneous, yes, it should follow the Poisson count distribution. Even, if inhomogeneous Poisson, so like you have a spike in demand that's very large, and whose timescale is much smaller than your interval window, ie 20x base rate for 15 minutes following lunch, while your intervals are 1 hr, your counts should still be Poisson. That is because the sum of Poisson counts is also Poisson, so Poisson(20x1) + Poisson(1)+Poisson(1)+Poisson(1) is still a Poisson.
However, if your process is inhomogeneous and history dependent, ie 1 taxi ride tends to increase the probability of the next taxi ride, for example let's say rider twitters that the taxi driver gave her a 75% discount, and everyone in the city responds and starts hailing rides, and they also get 75% discounts which they then twitter, your process is self-excitatory, and will probably have higher variance than Poisson, and would be better modeled as something like a negative binomial distribution? On the flip side, if she twittered and said the ride cost 400% of normal, leading to self-inhibition that might cause a time-dependent drop in ridership, probably leading to lower variance than Poisson, ie a binomial distribution might be better suited.