Is English the common language that unites them? Even though UK exited
Mostly, but there are also regional lingua francas. French in countries surrounding France. German in countries surrounding Germany and much of Western part of Eastern Europe, but English is mostly used as the main lingua franca. Nobody really views English as a British thing, it is seen more as a global business science standard language if you know what I saying.
The EU works a lot like the US, but on less effective level. Most important things that impact the daily life of EU citizens is managed on a national level (education, social welfare, etc.), while a lot of the business/product standards/trade treaties/environmental/higher level stuff is managed at the EU level (similar to the Federal level of the USA).