No, Generation Y is what the Millennials were originally called by demographers before the generation got old enough to call themselves something and marketers got on board and ran with Millennials.
There is a Millennial obsession with creating ever-smaller cohorts ("Xennials," "elder Millennials," etc.), but Generation Y was a demography thing. I remember it very clearly in Time magazine, in an article about how today's kids (Generation X in this case) were lazy, had no work ethic, were going to inherit a poorer world than their parents had, etc. All the stuff that was then recycled and the Millennials were hit with and Generation Z will get slammed with soon.
The great hope for the angry Baby Boomers and Greatest Generation readers (and writers) of Time magazine at that point was the new, better generation, Generation Y, which would be coming of age right around, you guessed it, the turn of the millennium.
(Incidentally, Generation X is also a name that the generation got stuck with. It was
popularized by the once-rising star novelist Douglas Coupland, who was everywhere in the early 1990s.)