The general principle is that voting rights are based on citizenship and each country makes its own rules. When electoral rights are granted to non-nationals, these are usually limited to local elections and do not extend to national ones. So neither EU nationals or non-EU citizens are able to vote for example in French presidential elections or German parliamentary elections, unless of course they have taken citizenship in those countries.
Moving abroad for tax savings and an elevated lifestyle is appealing to many Americans, but understanding each country’s rules and tradeoffs is crucial before relocating.