A nanny is usually a W-2 household employee.
See IRS Pub 926 page 3.
https://www.irs.gov/pub/irs-pdf/p926.pdf
Absolutely not. Publication 926 (2020), Household Employer's Tax Guide ...
A nanny does not meet the requirements for self-employment. A nanny does not choose when to work and other requirements. See Understanding Employee vs. Contractor Designation | Internal .