Many state payroll taxes are considered state income taxes for the purposes of the federal itemized deductions. See 2020 Form 1040 Schedule A instructions for line 5a:
If you don't elect to deduct general sales taxes, include on line 5a the state and local income taxes listed next.
Mandatory contributions you made to the California, New Jersey, or New York Nonoccupational Disability Benefit Fund, Rhode Island Temporary Disability Benefit Fund, or Washington State Supplemental Workmen's Compensation Fund.
Mandatory contributions to the Alaska, California, New Jersey, or Pennsylvania state unemployment fund.
Mandatory contributions to state family leave programs, such as the New Jersey Family Leave Insurance (FLI) program and the California Paid Family Leave program.
This doesn't address Washington state's long-term care tax specifically, probably because that is a new tax that only starts in 2022, but based on how the deduction applies to similar existing taxes, it would seem like yes, it should count as state income taxes for the purposes of the federal itemized deductions.