If you're not asked in your state interview for your state tax ID number, most likely your Federal Tax ID number is sufficient to identify you for filing your state taxes. You could check your state's DOR for more info.
You're correct that Schedule H only requires the Federal ID number. Be sure to enter your taxes paid in Other Tax Situations > Additional Tax Payments > Nanny and Household Employee Tax.
Here's more detailed info on Schedule H.
@jiminrb
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