"I think it does not matter who received the payment."

Put it on their 2020 return. That's all you do, now. Ignore who got it, why they got it, or when they got it. None of that matters, now.

"What the matter is "whether the money was dibursed from IRS due to the child being claimed." "

Not really. It was a Projection. They used 2018 or 2019 tax returns. That's why this does not matter Now.

"According to you a child's rebate can be claimed twice, one time when, for example, when mother claimed the child in 2019 return"

That's correct; the Congress, the US Treasury and the IRS had a stupid concept that has proven flawed. You don't control this.

"and then when grand mother claims the same child in her 2020 return"

Yes, this is now 2020 tax return Reality, and that is your responsibility = to report the reality. For the grandmas as taxpayer, she claims the EIP as a credit because she did not get Funds in advance for this child.

If there is a dependent claimed in 2020 and that is a Qualifying dependent for purposes of the EIP (which is different than other qualifications in the tax code, such as EITC, CTC, support tests, etc), and the Tax payer claiming the dependent has not already gotten the funds or the right amount of funds for 2020, then the refundable Credit is allowed and applied for.