After you file

It is not clear here what kind of confusion went on between you and someone else who wants to claim a dependent.   If someone claimed a dependent that YOU could claim, and your own e-file was rejected, you can file the return by mail and let the IRS sort it out.   They eventually send letters to both of you and you have to show the IRS who the dependent lived with.

No, that other person cannot "make up" for a mistake now by claiming a child for 2024 if the child did not live with them in 2024.   But they can file or amend a 2023 tax return to claim the child if a mistake was made; and if YOU made a mistake in regard to claiming a child you should not have claimed, then you can amend your return.

IRS interview to help determine who can be claimed:

https://www.irs.gov/help/ita/who-can-i-claim-as-a-dependent

https://ttlc.intuit.com/questions/3113432-who-can-i-claim-as-my-dependent  

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, or fiancé (etc.) as a dependent on your 2023 tax return as long as they meet the following requirements:

Qualifying child

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
• No age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

Qualifying relative

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $5050 in 2024.
• You provided more than half of their financial support.