Copa cup seemed like a more physical brand of soccer than Euro Cup. Is it a philosophy? Difference in talent? Different type of athlete? Or am I just wrong?
I think two reasons for differences you have seen are:
1) Format of each tournament
In the group stages of the Euros, 4 of the 3rd place teams qualified for the knockout stage, not so in the Copa. Some teams in the Euros were just looking to get 3 or 4 points, and didn't need to push for wins, where the need to win was more important in the Copa
2) Familiarity/Dislike
Every four years, for South American (CONMEBOL) WC qualifying, all 10 countries are grouped together and play home-and-home competitive games against each other. In addition, Copa is played more frequently than the Euros. The SA teams know each other well, and there are certain heated rivalries. With the larger Euro pool, teams may have likely never (or have rarely) played competitive games against each other.
These could explain the more physical games you witnessed in the Copa