This is quite tricky.
For one thing, your child is considered to live with their parent "at home" if their absence is considered temporary. Temporary absences include being away at college, so long as the child returns for school breaks, or leaves their stuff at the house, or it is reasonable that the child will return home after college while they figure out the next step in life. So, depending on the exact circumstances, this might be a temporary absence and you can say your child lived with you all year. From there, it will depend on the child's age and whether or not they provided more than half their own support.
If this was not a temporary absence, then you might be able to claim the child as your dependent even though they did not live in your home, if you provided more than half their total financial support. Or, you could claim the child if you provided at least 10% of the child's support and you get a signed statement from everyone else who paid more than 10% of the child's support that you can claim them and they won't. This works because you can claim your child as a dependent under the support test even if the child does not live at home, but this kind of dependent is called a "qualifying relative" and is worth a $500 credit, and is not eligible for the larger child tax credit, and does not allow you to file as head of household.
The other family can never claim your child no matter how much financial support they provided since the child did not live in their home all 365 days of the year.
https://www.irs.gov/publications/p501#en_US_2023_publink1000221013
