There are a couple of possibilities here for what is going on.
First, Crane wrote his book in 1895. The Rutgers vs. Princeton game that is usually seen as the first college football game happened in 1869. It would likely have been hard to look up the exact date for the game back in those days and Crane might have just missed by a few years.
Second, the 1869 game was not the first time anything called football had been played -- it was just the first with these particular rules that gave rise to what we now call football in the US. So Crane might have been referring to one of these other types of football.