7 countries won medals

6 Germany
3 France
3 Great Britain
3 USA
1 Sweden
1 Australia
1 Canada

Or
13 EU
3 USA
1 Australia
1 Canada

Quotas of horse/rider pairs vary between Games and between each discipline. Currently, each National Federation may enter a team of 4 riders on the jumping team (one of which is a reserve), 5 on the event team (no reserves), and 3 riders on the dressage team.

https://en.wikipedia.org/wiki/Equestrian_at_the_Summer_Olympics

These were the events in Rio:
Individual dressage
Team dressage
Individual eventing
Team eventing
Individual jumping
Team jumping

So if 1 country were to medal all their possible entries 1) team jumping 2) team event 3) team dressage 4) individual jumping 5) individual event 6) individual dressage; they would only be able to earn 6 medals out of 18 available...Germany apparently did exactly that.

If you further consider that the USA is #4 all time in Equestrian medals in Olympic competition behind 1) Germany 2) Sweden 3) France (all EU members) it is very possible that if the EU were one country then the USA could have also won 6 medals as well. So dropping the EU from 13 down to 6 and bumping the USA from 3 up to 6 would be a real possibility.