I meant that, negotiation aside, here it literally is their business (via form P45) in setting you up on payroll. (Unless you time it or take a break such that the new job is your only one since 6 April, i.e. in the current tax year.)
Thinking about it - I do know/can infer a little about the US - everyone does (or has someone do) their own taxes there, so it's not needed. The purpose of this in the UK is that the second employer within a tax year can set up the new employee correctly so tax isn't underpaid as a result of ignoring the previous income.
(PAYE - Pay As You Earn - income tax, national insurance contributions, and student loan repayments are handled by employer. People only 'do taxes' (self-assessment) here if self-employed, have capital gains over the allowance or sale price over the reportable amount, etc. extra curricular stuff like that that, especially because of allowances before the tax kicks in, most people don't need to. I would guess the most common reason is pension contributions made with post-tax money (employer's pension scheme is of course paid into primarily through payroll, not that) for those earning enough to have a higher marginal rate of income tax - you get the basic rate back on contributions automatically, since everyone pays it, but any extra requires self- assessment.)