I am always interested in this. I was involved in the campaigning against the re-introduction of ID cards by the Blair government, and I have family who lived just off Ballards Lane in Finchley, North London, scene of Clarence Willcock's refusal to show his ID card to a policeman in 1950.

Like the doomed 2000s proposals, the 1939 registration scheme was in two parts. A card was issued to individuals, but the card was really only a key into the national register. So a forged card would pass a cursory examination, because it would just be given a "does this look valid", but could be detected (or not, vide infra) because there would be no matching entry on the national register. The cards were, in principle, required in order to access rationed goods, which included food and clothing. We will also see below why this was clearly not effective.

The first reason these cards were less of a barrier than might be thought is because they did not carry a photograph. It is always a surprise to American car hire companies that there are still a lot of UK driving licences which do not have photographs: if you passed your driving test before, and have not moved house since, 1998 then your driving licence does not contain a photograph, and up until 1998 no driving licences did (edit to add: mainland England, Wales and Scotland; it was different in Northern Ireland). The same was true of other documents, and clearly in the hurry of late 1939 demanding photographs from everyone would have been completely impractical.

The second reason was that actually checking that a card was valid would have been hugely more difficult in 1940, or even 1950, than it might be today. To contact a central register and have them access handwritten paper records to determine whether card 12345 in the name of John Smith was valid would require telephones and manpower which were not available. So the cards stood on their own, so were (a) forgeable and (b) theftprone, as they were neither tied to a register for authenticity nor tied to an individual for identity.

That pretty much answers the OP's question, and even by 1947, this was well understood. To quote W S Morrison MP, speaking in parliament in 1947 in opposition to the continued renewal of "emergency" legislation

"Law-abiding citizens who live in one community are particularly prone to lose them because they are known by all their neighbours and do not carry the cards. The dishonest man—the spiv, as he has been called—is generally possessed, I am told, of five or six different identity cards which he produces at his pleasure to meet the changing exigencies of his adventurous career. So in the detection and prevention of crime no case can be made out for the identity card. I understand that this is also a matter which is being touched upon in the Criminal Justice Bill, but I offer it as my opinion that the unjust are possessed of many identity cards whereas the just are frequently placed in positions of embarrassment by temporarily losing them.

The argument advanced on Second Reading—I conceive it to be the main argument for the retention of these troublesome documents—was that as long as rationing persists they are necessary. I do not believe it. We were told in the House the other day that there are 20,000 deserters still at large. How have those 20,000 persons contrived to equip themselves with food and clothing? Ex hypothesi they cannot be possessed of valid, honest identity cards, but that has not prevented them from sustaining themselves with food and clothing themselves with raiment without those documents."

These arguments were made, pretty much unchanged, by conservative MPs in the noughties. The police vaguely said that ID cards would help in the detection and prevention of crime, but would neither accept increased targets (after all, they'd have a new tool to prevent and detect crime, so should be able to do more with the same resources, right?) nor reduced budgets (after all, they'd have a new tool to prevent and detect crime, so should be able to do the same with fewer resources, right?). So if it neither reduces crime nor saves money, what's the point? The same applied, mutatus mutandis, to benefit fraud: it turned out, under close examination, that very little benefit fraud was about mis-representing your identity, but it was mostly about mis-representing your circumstances, about which an ID card will do very little.

That said, the ghost of the 1940s scheme lived on and lives on. What is now called the NHS Patient Demographic Service inherited the 1939 register, and as the 1931 Census was destroyed during the war and the 1941 Census for obvious reasons did not happen, the 1939 register was the best starting point for a national health scheme. I cannot this moment remember where the 1939 register was based, but it was somewhere incongruous like Scarborough; the various arms of the NHS that deal with patient records have remained in the north of England, and whatever the HSCIC (Health and Social Care Information Centre) is currently called is based in Leeds. The 7000 volumes of the survey are now largely available to family tree enthusiasts.

The NHS number which, maddeningly, is not the same as your National Insurance number, was until 1996 in the same format as the wartime ID card numbers. For people who had been issued an ID card before their abolition in 1952 their NHS number actually was their ID card number.

So the tl;dr is "they were relatively easy to obtain illicitly, either as forgeries or as stolen cards, and only provided very weak authentication".