Looking in to it i believe Mainland Europe and the Western World has nine countries without any form of national ID cards:
• Andorra
• Australia
• Canada
• Denmark
• Ireland
• New Zealand
• Russia
• United Kingdom
• United States Of America
ID cards are much more widespread that i thought.
It is estimated around 95% to 99% of adults in Russia have a passport. You need a passport in Russia to do anything. So for them ID cards are not required as everyone has a passport and people just use their passport for everything.
Andorra is in a very similar situation to the UK in that they have no alternative ID cards available other than driving licences.
Ireland and the USA both have a passport card which is kind of the equivalent to an ID card that most other countries have. The passport card in Ireland only allows you to travel to Schengen countries. The passport card in the USA only allows you to travel to Canada and a couple Caribbean countries. But only citizens of Ireland or the USA are eligible for one.
Canada and the USA both issue ID cards through their driver licence and motor vehicle agencies in most of their states and provinces. These are basically in the format of driving licences but say "identification card" rather than "driving licence" on them. I think any resident can apply for these.
Denmark has ID cards issued through each of their municipalities. This is not a national ID card as each municipality is responsible and has their own design and format. Also i do not believe Faroe Islands or Greenland have any so only Mainland Denmark municipalities issue them. They are not valid for international travel. Any resident can apply for one.
Australia has "Proof Of Age" cards which are available through the government of each state and territory. They can be used as ID cards. These are normally issued by their driver licence and motor vehicle agencies like Canada and the USA does.
New Zealand has a "Kiwi Access Card" which is issued by a private company. Any resident can apply for one. I think it is kind of similar to our "Citizen Card" in the UK in that it is issued by a private company and not really an official ID card and not everyone will recognise or accept it.
So really only four countries in Mainland Europe and the Western World exist without any official ID cards:
• Andorra
• New Zealand
• Russia
• United Kingdom
Albania, Austria, Belarus, Belgium, Bosnia & Herzegovina, Bulgaria, Croatia, Cyprus, Czech Republic, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Italy, Kosovo, Latvia, Liechtenstein, Lithuania, Luxembourg, Macedonia, Malta, Moldova, Monaco, Montenegro, Netherlands, Norway, Poland, Portugal, Romania, San Marino, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey, Ukraine, all appear to have national ID cards. However i think even all of these let you chose between a passport or an ID card. Even in the countries with mandatory ID you can still chose to have a passport instead of an ID card.
I do not think that any country actually exists in Mainland Europe or the Western World where you have to have an ID card and a passport is not an accepted alternative to having an ID card.