Multiple observations from the same individual are by definition not independent, so you would be violating the assumption of independence of the observations. Contrarily to what suggested by @ERT I can't think of observations that may be both from the same individual and independent. It seems a contradiction in terms to me. Even if multiple observations from different individuals were to follow distributions with exactly the same parameters, they would still depend from the individual that they are linked to (hence not independent) - individuals that simply happen to have exactly the same personality. Such a scenario is unlikely to emerge in reality but it could be simulated. I agree with @ERT in making driver-ID a categorical explanatory variable, but you should use it to analyse your data in a mixed-effect framework, by fitting driver-ID as a random effect. Yours is a classic example of data that require a mixed-effect model approach, which was developed (also) to analyse data that violate the assumption of independence (examples: repeated measurements on the same individuals, multiple flowers from the same plant, etc). See Pinheiro & Bates's "Mixed-Effects Models in S and S-PLUS" (2000) but also here and here for reference. Pinheiro & Bates's book, as well as this one by Zuur et al. provide further details on non-independent data and how to analyse them in R (including examples similar to your scenario).
For example, in R:
model1 <- glmer(number.hard.brakes ~ trip.length * driver.age + (1|driver_ID), family=“poisson”, data=YOURDATA)