I can only speak to the U.S., but....
You're correct, voter registration is a substitute of sorts for needing a government ID. You register to vote in the US to establish both that you are eligible to vote (e.g., 18+ years old, not an illegal immigrant, etc.) and also establish your residency so the election board knows who you should vote for; that is, what congressional district you live in, what local government officials you should vote for, etc. U.S. Citizens vote for a myriad of candidates in local, state, and national election races, and depending on where you live, the contest in which you vote changes. The problem with automatically registering all voters is that there is no mechanism by which the government knows where your residence is, so you register to tell the government where you're qualified to vote.
You're not required to have an ID card, but it makes life a lot easier if you do. Citizens aren't issued an ID card, but are issued a number by the Social Security Administration (aptly named their Social Security Number) and that number is used for a lot of ID-related items. If the police wanted to check the citizenship status of someone the SSN would help establish that. This is also the number that you would use to file your tax return, open a bank account, etc.
Court summons need to be delivered in person in most jurisdictions. This is done by the local sheriff, or a process server. Obviously they try to serve at a last known address, but sometimes it requires a little bit more research.
Demographic info is captured by a public census that occurs every 10 years. The Census is actually required in the U.S. Constitution. They do other quantitative research on people in between, but the Census is the only official record of resident statistics.
Residence in a home owned by the resident is established by a title deed that is filed with the local county government, or a rental residence is established by a lease agreement. Either are required to obtain a state-issued ID card, and most people have a state ID. If you didn't for whatever reason, you'd use the title deed to the home you own or lease to the apartment you live in to prove your residence.
The interesting thing is that in the U.S. there is a cost for obtaining a state-issued ID card, in addition to the paperwork necessary to receive one. So voters are not required to show an ID card when they vote in some jurisdictions; doing so is argued to be a form of voter disenfranchisement: if you can't afford the cost of the ID, or can't spend time away from work to go to the government office that issues the ID to get one, and aren't allowed to vote as a result, you've essentially been denied your right to vote by the government, which is a big no-no. Plus, election boards use other ways to verify identity besides looking at an ID card to prevent voter fraud (like requiring you to match signatures to your voter registration, or verifying the voter's address when they go to vote), so an ID isn't necessary.
I'm not sure that the lack of an ID card or voter registration is the reason for low voter turnout. Registering to vote is stupid-simple in the U.S.: you fill out a form (online, or mailed in), and wammo!, you're registered. The reason for low voter turnout is voter apathy--people who don't think voting is important enough relative to the effort to go out and vote.