In order for something to be a "generational anthem" it has to go beyond your typical love song or rock ballad; its got to make a statement about the era:
Silent Generation (1924-35): Somewhere Over the Rainbow by Judy Garland (1939). I would argue this is the most significant pop performance of the 20th Century, and the song almost perfectly encapsulates the mix of hope and anxiety that defined the Great Depression and its children.
War Babies (1936-44): Johnny B. Goode by Chuck Berry (1958). This may be the most recognizable rock song of all time, and I think its a natural fit for the teenagers of the 1950s.
Baby Boomers (1945-54): (I Can't Get No) Satisfaction by the Rolling Stones (1965). So many good choices here:
My Generation,
Imagine,
Like a Rolling Stone, but I had to go with the one that's about sexual frustration and commercialism.
Gen Jones (1955-64): Material Girl by Madonna (1984). What could be a more appropriate anthem for the conspicuously consuming yuppie generation?
Generation X (1965-76): Smells Like Teen Spirit by Nirvana (1991). This was the easiest choice on the list. Cobain's anti-anthem is perfect for a generation searching for authenticity amid material cynicism.
Generation Y (1977-85): Ride by The Vines (2004). The 2000s indie rock sound defined this generation, and what song could better represent the tech-obsessed Gen Y'ers than the one that got famous from the first iTunes commercial?
Millennials (1986-96): Ni**as in Paris by Jay-Z and Kanye West (2011). This collab is the epitome of 2000s hip hop materialism performed by the two biggest R&B artists of the era. I thought of several other possibilities, but settled here.
Homelanders (1997-05): Royals by Lorde (2012). This song is a reaction against the glitzy, bejeweled lifestyles of the Millennials' hip hop icons. I think it speaks to the material anxieties of the first entirely post-9/11 generation and children of the "Great Recession".
Zoomers (2005-2015?): It's still too early to really, but I'll go with
Old Town Road by Lil Nas X (2019). What could be more Zoomer than an LGBT POC independently releasing a country/rap single on SoundCloud?
There are some glaring problems with this list. For starters, the 1970s (perhaps the best decade for American pop music) is entirely missing from this list and some of the anthems run a bit too close together but, hey, generations are completely made-up and arbitrary anyway.