At what rate is severance pay taxed by the IRS to the employee - the tax rate of the employee or the 22% withholding rate of the employer?

It's taxed the same as any other ordinary income. So for your client, looks like the highest bracket and the 22% withholding is not enough.

Ask the employer to withhold extra or pay in an estimated tax payment right after he deposits the payroll check if that is a concern.

You asked "at what rate is severance taxed" and this is confusing Tax and withholding.

There are two Flat Rates for Withholding from "one time" or Bonus payouts. That has nothing to do with the taxpayer's entire life or their tax bracket. This is in Pub 15. There are three ways to compute it, though, depending on if it is combined with ordinary wages, paid on a separate check, etc. Mostly, people use the flat rates, since it is easier.

Up to $1m = 22%

Over that, jumps to 37%: "If a supplemental wage payment, together with other supplemental wage payments made to the employee during the calendar year, exceeds $1 million, the excess is subject to withholding at 37% (or the highest rate of income tax for the year). Withhold using the 37% rate without regard to the employee's Form W-4. In determining supplemental wages paid to the employee during the year, include payments from all businesses under common control."

https://www.irs.gov/publications/p15#en_US_2022_publink1000202353

This section is called Supplemental Wages.