If I didn't receive my 2nd stimulus check until mid January, am I still supposed to claim it on my 2020 taxes??

Yes, you need to file the 2020 federal tax return for the stimulus payment.

The stimulus questions are asked in the Federal Review section of the TurboTax program.  Any payment you were eligible to receive based on the 2020 Federal tax return but did not receive will be entered on the Federal tax return, Form 1040 Line 30 as a tax credit.  The credit is included in the federal tax refund or it will reduce the amount of federal taxes owed.

YES ... that is why the program asks for it.  Both stimulus payments were ADVANCES on the 2020 credit.  Failure to enter it on the return will delay processing while the IRS corrects what they are calling a "math error".