Has an Euros game ever ended 5-3?
It has. 5-3 has been a Euros final 6 times in 3,212 games. The first was Hungary beat Poland on May 17, 1987; the latest was Spain beat Croatia on June 28, 2021.
That makes 5-3 the 23rd most common final score out of 34 that have ever occurred in Euros. One point away, 6-3 has happened never and 5-4 has happened 1 times.