At least according to exit polling (which are not gospel on this sort of thing but generally backed up by poll subsamples at the time so reasonably believable), Bush narrowly won 18-25-year-olds in 2000. Thus, they were marginally more Republican than the nation as a whole, as Gore overall won the popular vote very narrowly. There is generally no evidence that younger voters were voting more for the Democrats than older voters in the 90s or earlier as well.
If you want to go by how much more Democratic they voted compared to the nation, then sure I suppose in 1992/1996 you could say it was rather unremarkable, but I don't really see it that way. How voters vote in their youth can be highly suggestive of their future, and in Clinton's case, those voters have displayed modest Democratic leanings even as they aged. Bill Clinton was a popular president who presided over a booming economy, so the idea that young people growing up under him may have taken a shine to Democrats is not far-fetched. All presidents, good or bad, have an effect on the young people who grow up under their tenure.
What you said about Bush winning some portions of younger voters, sure, I think that could
suggest that Clinton's impact among 90s youth was more isolated to him and the circumstances of the time, and was not really connected to the main trend that caused young people to begin heavily trending Democratic in 2004-on.