Any where in the world is not the same as ones home country. Any country in the world is not the same when living there as when passing through as a visitor.
As a very general way of making an initial choice, what equestrian disciplines are you interested in? The American Hunter-thing doesn’t exist in Europe.
Showjumping then Belgium and the Netherlands have top shows every week. The distances are so small you could be from one side of Belgium to the other in about two sneezes.
Dressage, then Germany, Netherlands, Denmark are probably the tops.
Eventing, then the UK
Endurance, France
Western is very niche but France, Italy and Germany probably have the most activity.
Many EU countries actively promote equestrian sport and the horse is deemed to be an agricultural animal. The UK, of course, is different: very little official support. Such subtleties can impact e.g. access to club stables and other facilities, training and licensing for professional horsemen, structure for amateur riders (in France, for example, an amateur must meet minimum requirements and be licensed to show), how much access there is to the countryside to ride… Such things take a lot of research and local knowledge.
One consideration is that the South/Mediterranean Europe doesn’t have grass like northern/Atlantic Europe. The best hay I have ever seen was in Finland.
Northern Europe has long, dark winters which are longer and darker the further north one goes. The British Isles are further north than the USA and the big Canadian cities.
Land is expensive: populations are large and dense and history prevents urban sprawl.
Travel with horses within the EU is very easy, passports for horses being accepted in other states, veterinary requirements being standardized. The exception is the UK - now outside the EU so costs to travel a horse to compete in Europe have more than trebled and the bureaucracy is a nightmare.
There is great variation within the larger nations: the south of France or Germany is very different from the north, for example.