There are countries where each citizen has one unique identifier (Sweden's "personnummer", Denmark's CPR).

The UK is definitely not one of those! [yet]

Instead there are many different identifiers, each for a different purpose, and stored in different systems which almost certainly don't talk to each other.

Just for starters: NHS number for healthcare, National Insurance number for social security and pensions, Unique Taxpayer Reference for tax, Passport (with a number that changes when you renew your passport), Driving licence (with a "number"[alphanumeric] which stays constant even when you renew)...

Multiple overlapping identifiers... and I may have missed some :)

There is, but it's not tied to any strong identity verification process, and so there's a thriving fraud where unemployed citizens will rent out their numbers to working illegals. It's not something that the tax office has ever really worried about, since if anything it tends to increase the amount of tax paid (if several people are sharing the same tax ID they'll pay a higher tax rate), and while they might bat an eye at someone with 5 different salaried jobs it's not particularly suspicious when it's gig economy work.

The wrinkle is that it doesn't seem to be tied well to identity. Someone working illegally can provide an NI number that's legit but not theirs. Their work accrues to someone else's NI record, but the person getting the extra years probably never notices and the person working under their NI number doesn't care because they aren't entitled to a state pension anyway, they just want to work now.

Here’s what employers need to do currently: https://www.gov.uk/check-job-applicant-right-to-work