Here is something which sounds like an entry in a You Know You're Getting Old When list, so I might as well phrase it like that: You know you're getting old when the stuff you consider "classic" is just "old" or (polite way to put it) "old-school" to everyone else.
I was born in 1971, and immediately started doing everything in my power to fix that sorry state of affairs. To that end, I have a slightly skewed idea of what constitutes old-school anything.
But since the original conversation revolved around anime, my rundown of old-school in that department went something like Gatchaman / Battle of the Planets (depending on where you came in), Galaxy Express 999 and most of Leiji Matsumoto's work in general, Ikkyu-san (which aired on UHF in Jersey), etc. Other people chimed in with "old-school" titles that for me were the cusp of my involvement with anime generally: Utena, Lodoss Wars, Bubblegum Crisis, Akira.
Several things become clear from all this:
The cutoff for old vs. new is always personal.
It also doesn't have to be tied to one's personal experiences, but rather one's perceptions of personal experience. The stuff I was watching as "old-school" had already been out for a good long time; it wasn't "new" when I saw it — and I think I had some perception of that.
Calling something "old-school" isn't always dismissive or derogatory, although it can come off that way to people who think of such things as roots or essential. They feel those things need to be seen to understand the presence of current things, but more recent viewers aren't, by and large, scholars or cultural archaeologists — and it's a little unfair to suggest their enjoyment is going to be that much shallower if they don't take up such a role.
This last point's the most crucial one: Most people don't care where something came from. They don't care that Xerox PARC invented the mouse or the windowing system; they just care who's currently doing it right. Likewise, they don't care what the roots of this shonen trope or that character type came from; they just care about the most current incarnations of it. Archaeology is for nerds.
It's one of those things that every "second-order" fan — fans who write about, critique or otherwise go that extra mile with their fandom — has to make their peace with. Otherwise, you just make yourself feel like it's you vs. a universe of uncaring dolts, when the reality is far less adversarial.