This varies country by country. I would say roughly half the countries require one and the other half not (and even less require it for foreign citizens)
There is some other weirdness though. As we are in the Schengen area there is no border control and thus the police have been given the authority to look for people here illegally and thus they are allowed to ask anyone to identify themselves basically without any reason. If you can't identify yourself they can take you to the police station to verify who you are.
https://en.wikipedia.org/wiki/National_identity_cards_in_the...
And in some places (like Finland) you can vote without one if you can prove who you are in some other way (for example have a relative with you who has one and the police give out temporary free ID cards for voting). Though as the drivers license works as a valid id in Finland it usually isn't much of an issue (very few actually have the official id card thingie).
Also everyone registered to vote automatically. For early voting you can use any polling station (on the actual election day you have to use the one assigned to you)
No, but it is nontheless relevant: in the Netherlands one is compelled to carry ID. Absolutely no such requirement exists in the US. So it's a different situation where an ID is required to vote in a country where carrying ID is already compelled compared to requiring an ID to vote in a country where no such ID-carrying requirements exist and many find the notion of compelled ID-carrying odious.
A marriage also isn’t as time-sensitive as a vote, so that’s somewhat different. If someone who lacked ID goes to the poll to vote and is rejected for not having ID, this is different from going to the City Hall to register a marriage. You can do the marriage on another day. The vote, not so much.
It is also a hangover from Nazi occupation. The UK has no ID that you have to carry at all times, nor does the US.
So where as requiring ID in the Netherlands is a non issue it's a massive issue in the US, especially as many people don't need a passport if they never leave the US.
Personally I prefer the US and UK systems. I like the basic level of anonymity that you have from not having to carry ID.