No, you would not both fully claim that income.
Is this a Personal Residence or Rental.
If Personal Residence, the "Rebate" is only taxable if Itemized Deductions were used the year before.
Itemized Deductions allows you to deduct taxes. If you deduct taxes one year, but get them "Refunded" or "Rebated" the next year, you need to claim the excess as income.
If you both used the Standard Deduction for your 2021 Tax Year Return, neither of you claim that income.
If one or both of you claimed taxes paid as a deduction on your 2021 return, that person will need to adjust for the rebate. The rebate would be taxable to the person that claimed the tax paid last year up to the amount they claimed. If you each claimed half for 2021, you would each claim half the rebate for 2022.
If the property is a rental, the rebate is income.
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