If the EU were its own country they would have had less medals in total because of a few reasons.

There is a limitation of the number of athletes you can send (which has already been stated). This is a real issue, and yes I get the argument that they still would have won. But in sports in which your country is dominant, some of the best athletes are left behind because of the limitation of team size.

Also you have to factor in team sports. For example, in some sports that are dominated by EU members (rowing, cycling, equestrian, etc) instead of getting the gold, silver and bronze (it's likely you only get one). In team relay competitions, you would only get one medal and not multiple. In team sports like basketball and volleyball, you would only get one medal instead of multiple. In Team Handball, EU countries took all three medals. If the EU were its own country they only get one. This scenario would have been played out over and over again.

The EU as a country would have had more medals but not 400 like they claimed. I think they would have had more than the United States, but what if we added Canada and Puerto Rico?