"Are you really saying that formerly anyone could go into a polling station in the UK and state that she was Mrs Jane Smith of 10 Main Street or Mr John Brown of 19 Bank Street and cast a vote without anyone asking for their social security number?"
In a word: Yes. Those registered to vote are sent Polling Cards through the post, with all their details included, but there is no necessity to bring them along with them, nor to show them.
My (English) district council took part in the trial a couple of years ago, requiring voters to show ID, and I did Presiding Officer duties at my local polling station. There was absolutely overwhelming support for voter ID amongst those who came to vote - and the voter turnout was as normal.