Has an Euros game ever ended 5-3?
Exactly once. On June 28, 2021, Spain beat Croatia 5-3, and no Euros game before or since has ended that way. When it happened it was a scorigami, the first 5-3 in league history.
That makes 5-3 the 15th most common final score out of 20 that have ever occurred in Euros. One point away, 6-3 has happened never and 5-4 has happened 1 times.