In my experience, it's the opposite: older Millennials are self-identifying as GenXers, and younger Millennials are self-identifying as Zoomers. Because that's how bad the stigma against Gen Y has become.

Gen X and Gen Z are being portrayed by the media and pop culture as "cool" and "enlightened." Gen Y, sandwiched in-between those two cohorts, are viewed as the entitled misfits. No one wants to be considered part of our cohort. Everyone takes shots at us -- because, apparently, there always has to be a Cheese_Who_Stands_Alone in society.

Of course, everything you're saying about the similar ageist stigmas against Baby Boomers is also 100% happening to your generation, as well. I think it's fair to say that Boomers and Millennials are tied as being the two most unfairly-stigmatized generations of the modern age. It's become "fashionable" to dump on both.