The taxes withheld on the 2021 unemployment (along with all the unemployment income) will be reported as payments on the 2021 return and any excess withholding becomes part of the refund on the 2021 return.
The 2022 "right of claim" will reduce the taxes you owe on the 2022 return period. If you will get a refund on the 2022 return depends on all your other 2022 income, credits and withholdings.
