Id say it’s not a difference in the politicians but the citizens. Pessimism and paranoia are rampant in the UK. We already went through this ID card debate 20 years ago and the fear-mongering won. So the idea just reignites that debate with a lot of baggage.
The UK has various systems in place to ensure people are legally allowed to work, rent, etc but in reality they inconvenience people without actually catching “the bad guys”. This system would make life more convenient and make the chance of catching the bad guys higher.
In truth though the problem is dodgy employers on a large scale. Take Deliveroo or Uber Eats. The accounts are rented out to illegal workers. You could literally catch one for every order you make. But for some reason the government isn’t actually going after the obvious hanging fruit.