So the eligibility language for the CA EITC Dependent credit requires an SSN or ITIN:
https://www.ftb.ca.gov/file/personal/credits/california-earned-income-tax-credit.html
You must:
- Have taxable earned income
- Have a valid social security number or individual taxpayer identification number (ITIN) for you, your spouse, and any qualifying children
- Not use “married/RDP filing separate” if married
- Live in California for more than half the year
So to be eligible for the credit you and the applicable child must have an SSN or ITIN.
It probably goes without saying that I'm just a stranger on the internet and don't know anything and you should talk to a tax professional, but I suspect there's no point to claiming the FTB is somehow illegally using the ITIN.
On the IRS page related to the ITIN, immediately after the part you quoted, the IRS expands:
An ITIN does not:
- Authorize work in the U.S.
- Provide eligibility for Social Security benefits
- Qualify a dependent for Earned Income Tax Credit Purposes
So the IRS is saying the ITIN is only for tax tracking and doesn't authorize any other thing; it's not saying it's illegal for anyone to ever request or use your ITIN. Sort of like the way my drivers license says something like "this doesn't establish eligibility to work or vote." But again, talk to a tax person, not a stranger on the internet.