I received the second stimulus check in January, 2021. Does it legally belong to the tax records of 2020?

Yes a payment you got in January or February 2021 is an advance and based on your 2020 return. Enter the second Stimulus payment on your 2020 return now. The year you got it doesn't matter. If you don't report it Turbo Tax will give it to you again, which would be wrong and double. The Stimulus entry screen even says the second one is between Dec 2020 and Jan 2021 so you should include any payments you received in January and February.

Yes.  If you do not enter it you may delay your refund.

Congress passed the 2nd stimulus bill at the end of 2020. Both stimulus checks were an advance credit for your 2020 tax return,  The 2nd stimulus checks started going out at the very end of December 2020 and continued into early January 2021.  When you fill out the recovery rebate credit it is important for you to enter the entire total of the amount you have received for yourself, your spouse, and your dependent children so far.   Entering the wrong amount might delay your 2020 refund.

It should help you if you bear in mind that the purpose of the question is to determine if you have already received the amounts you should have received.