The moon always remains the same color as it always is. The “blue” may come from the Old English word “belewe,” which means “betray.”
It is the “extra” full moon because a year which has 12 full moons gets 13 instead.
Perhaps centuries ago, this was viewed as the moon “betraying” mankind or tricking people with a second full one.
The moon’s cycle is 29.5 days, while months can be as long as 31 days or a short as 28. That misalignment between the calendar and the moon’s schedule causes the doubling-up.
Because of this, the phrase “Once in a Blue Moon” is used to depict rare events.