It's a dreadful idea.
Hypothecated (i.e assigned to one specific purpose) taxes at a fixed single rate are 'regressive'; in other words the people with the lowest incomes pay out the largest percentage of their income.
It suggests that people who take out private insurance to cover care costs would be exempt. I can see this creating a two tier care home system straight away!
What happens to people who are unemployed? Are they credited with contributions? Do they get a lower tier of care because they missed contributions?
And what happens if the government of the day decides that there's not enough in the 'pot' and so lowers standards to save money?
The idea that our 'national insurance' contributions actually pay for our future needs is a myth, which I'm sure most people recognise. I suspect that a 'hypothecated tax' would be used in much the same way, for current needs, not the future needs of the person paying it.
We discussed this a bit on one of the recent NHS threads, but I think we really need to completely lose the idea that government spending is from 'taxpayers' money'; an idea which creates resentment about non-taxpayers because of perceptions that they are not contributing anything, but start thinking of money coming from the government as 'everybodies' money' to be used for the benefit of everybody, by consensual agreement.
As was said on the NHS thread, we, the nation, can afford to pay for anything we want to, our money creation powers are limited only by the danger of scarce resources causing inflation. Why not just create and finance a decent social care system. The money will return to the treasury eventually and it will grow the economy in the meantime.