California has a similar form, but you don't need to mail it or related documents to the state.
According to Part V – Declaration of Taxpayer(s) in the 2023 Instructions for Form FTB 8453, California e-file Return Authorization for Individuals, an e-filed tax return isn't considered complete or filed unless you sign Form FTB 8453 before your return is transmitted.
According to C, Taxpayer Responsibilities in the instructions:
After your return is e-filed, you should retain the following documents for the California statute of limitations period:
• Form FTB 8453 (signed original or copy of the form).
• Original federal Form(s) W-2, W-2G, and 1099-R.
• A paper copy of Form 540, California Resident Income Tax Return; Form 540 2EZ, California Resident Income Tax Return; or Form 540NR, California Nonresident or Part-Year Resident Income Tax Return.
• A paper copy of your federal tax return.
• A paper copy of your other state income tax return if you claimed the Other State Tax Credit. Refer to California Schedule S, Other State Tax Credit.
See the instructions for additional information.