Has a Copa America game ever ended 3-3?
It has. 3-3 has been a Copa America final 7 times in 869 games. The first was Bolivia and Colombia drew on February 21, 1945; the latest was Peru and Paraguay drew on July 2, 2021.
That makes 3-3 the 18th most common final score out of 36 that have ever occurred in Copa America. One point away, 4-3 has happened 7 times and 3-4 has happened never. At one point the score disappeared for 48 years, between 1945 and 1993.