Signing on behalf of your child, as suggested in the IRS instructions, would probably be sufficient for your state return, but you're correct. California's 2019 540 Forms and Instructions don't have a policy statement similar to, for example, what's in the Responsibility of parent paragraph in the IRS's Publication 17.
However, perhaps filing a Power of Attorney with your child's return will give you the assurance you want.
The California Form 540 instructions state:
"Power of Attorney.
If another person prepared your tax return, he or she is not automatically granted access to your tax information in future dealings with us. At some point, you may wish to designate someone to act on your behalf in matters related or unrelated to this tax return (e.g., an audit examination). To protect your privacy, you must submit to us a legal document called a “Power of Attorney” (POA) authorizing another person to discuss or receive personal information about your income tax records.
For more information, go to ftb.ca.gov/poa."
And the state's Help with power of attorney page gives guidance on filing a POA for a minor child.
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