The old house, when sold, will need to be reported for taxes. If you sell it in 2021, then it'll go on your 2021 taxes. If you sell in 2022, then it'll go on the 2022 taxes. Assuming you qualify for the exclusion on sale of primary home deduction, that shouldn't change for either year. If capital gains tax rates change between tax years, that may impact you.
The old house would just be a 2nd home. Not a rental, not investment property. As long as you don't sit on it for 5 years and lose the sale of primary home exclusion, you're fine.
Much more pertinent is the fact that you're carrying two houses. Two mortgages (if you have them), two sets of real estate taxes, who sets of utilities, two sets of exterior maintenance, etc. That adds up, quickly.