The school opened in 1709 as a 'charity school', meaning it was free to attend and the students would have been from poorer families (at the time, most well-to-do children would have been educated at home). The school was originally in the bell tower of the Trinity Church, but by the time you're seeing it, it's moved across the street.
This is the only school that remained open during the British occupation of New York, possibly because it was endowed by Trinity church, which was Loyalist. (Of course, Trinity also funded King's College, which was closed and used a military hospital, but universities are known liberal hotbeds.)
After the Revolutionary War, the government started to focus on free public education and the city eventually stopped funding church schools, preferring to fund secular ones. Trinity School became a private preparatory school, which it still is. It's prestigious, expensive, and it's been accepting students for 300 years. That's a long time... for the colonies. Of course, Eton was founded in England in 1440 - that's 20 years before Ezio was born, and around 50 years before Columbus discovered an America that Etonians would enjoy a proud tradition of looking down on for centuries to come.