One major difference in the political systems is that most European countries elect their parliaments with proportional representation (exceptions mainly UK and France), meaning voters vote for a party nationally, and then each party gets the exact percentage of the vote they got in parliament, so a party with 23% of the vote gets 23% of the seats. That leads to far more competition among parties than in for example the US, for example in Germany where I live there are 6 parties in parliament now, with between 10% and 33% of the seats, "the lesser of two evils" is not really a thing here. This increased competition leads to much less hostility between parties, but another important factor in that is that almost all European countries (except mainly France) are parliamentary republics, meaning the majority in parliament elects the government. You can't alienate half the parliament if you want to form a coalition.

Anyway, you said you want to look more closely into how the EU is run, I can recommend you a few things for that if you like. I personally have learned a lot about the problems of the EU from former finance minister of Greece Yanis Varoufakis during its financial crisis and European bailout. I have to warn you though, he says of himself that he'd like communism if it was politically feasable, and obviously having been a politician himself he may or may not have a stake in portraying his actions in a positive light, but here's basically the main things he's talking about, in an interview with a German political journalist I like