In this video, I mention doing some research, so here are the high level answers:
Columbus Day was first observed at this time because October 12th was the date Christopher Columbus arrived in the Americas. It has been observed for a long time, but was made a Federal Holiday in the United States in 1971, with a floating date that makes it always a Monday.
However, some 14 states and more than a hundred cities have officially voted to either not celebrate it, or have replaced it with Indigenous People’s Day. Some have argued that it should be a different day to avoid conflict, though others argue that replacing Columbus Day is deliberate as he should not be celebrated.
National Coming Out Day was first observed in 1988, and is on October 11th because it was the anniversary of he 1987 National March on Washington for Lesbian and Gay Rights. So the dates being in common was a coincidence on the basis that being a national holiday was probably part of why they had the march on that day, but there is no connection otherwise.
