Received 2022 1099-MISC for Payment received 2023

We received a 2022 1099-MISC from California Franchise Tax board (FTB) for the Middle Class Tax Refund (MCTR). However, we didn't receive the payment until (January) 2023.

This may be a mute point because the MCTR money may not be taxable on the federal return anyway (from what I read here)?

However, in general, how is this supposed to work when the issuer submits a 1099-MISC for payment issued at the end of year, but the recipient doesn't receive the payment until the following year? Can the Tax payer argue that the 1099-MISC is for the wrong year and declare it the following year instead...? How?
I tried to ask FTB to correct it (using Chat) but they said that it is the issue date that counts, not the date the recipients get the money.... Is that correct?