ID is not a requirement for democracy. ID requirements are a convenience for the state, and it's possible to run a society without them without being overrun by fraud.
It's useful for people to be able to present ID to receive entitlements, but it's only there because the state has imposed a requirement.
Nobody asks for an ID when you vote in the UK. You show up, tell them your address and name, they look up up in the book and give you the ballot. There's no check that you are who you say you are at all - you could be anyone.
Yet the UK is a functioning democracy. So ID certainly is not a requirement for democracy.
> You show up, tell them your address and name, they look up up in the book and give you the ballot.
When they talk about ID they don't mean a document you carry around with you.
They mean the processes that allow someone in England to register to vote, and then walk into a polling booth and give their name and address and get a ballot paper in return.
Using your example: you can't vote twice. You walking into your local station, and use your vote. If you return later in the day your identity is known to have already voted. If you walk to a different polling station your identity is not known there (even if you take your passport), so you can't vote there.
That is a good point, I hadn't thought about it like that. However the requirements to register to vote in the UK are pretty lax as well - to go on the electoral roll you tell the government your name and address, and you pinky promise that you're telling the truth. They then mail you polling cards for any election, which you don't need to use.
You don't need to give them any national insurance numbers or other ID to do this. It literally amounts to "I'm Joe Bloggs and I live at 20 Smith Street". That seems like the bare minimum required to have a voting system to me.
It's useful for people to be able to present ID to receive entitlements, but it's only there because the state has imposed a requirement.