Reading https://www.tax-rates.org/california/san-francisco-income-tax it seems to suggest that my employer is liable to pay a 1.50% tax on my income due to me living in San Francisco despite the offices being in another location of the Bay. Will they pass down that cost to me by deducting the corresponding amount from my paycheck or is that illegal?
The tax in question does not exist.
San Francisco does not have a local income tax. It used to have a 1.5 percent payroll tax, but that tax was repealed and replaced with a gross receipts tax in 2020.
Generally, though, an income tax is a tax on your income, and you are the only one receiving income in the transaction where your employer writes you a check.
The income tax is therefore your tax to pay, and the employer has no cost to pass down to you. It is merely obligated to withhold the tax from your paycheck and remit it to the government on your behalf, no different than the taxes that it withholds and remits to the IRS to satisfy your federal income tax obligations.
A payroll tax, on the other hand, is a tax on the employer's payroll, which the employer would not have been able to "pass down" to you, at least not directly by docking your pay. Indirectly, though, it is probably axiomatic that these taxes eventually get passed on to the employees one way or another. If the employer's costs go up, it is quite common for the employer insulate his own pay by not only increasing prices but by offering lower salaries, delaying raises, or reducing bonuses.
An "income tax" is levied on employees; it may be directly withheld by employers from the employee's gross wages or may be paid directly by employees, but in either case it's the employee's money being paid, and the requirement to pay is generally based on the employee's circumstances (in particular, where they live and where they work) rather than, say, where the employer is incorporated.
A tax which is levied on employers based on how much they're paying employees, and not reflected as a deduction from gross wages, is generally referred to as a "payroll tax". However, some taxes which are deducted from the employee's gross wages, such as Medicare and Social Security contributions, are also considered "payroll taxes".