Simple answer: if your boyfriend (BF) did not claim the child on his 2019 tax return, you may claim her on your 2019 tax return*.
That said, the statement "he claimed her on his stimulus" is, most likely, not correct. Anybody who files a tax return, every year, does NOT make a separate stimulus claim. The stimulus is paid automatically based on the most recent tax return. So, if your BF got the stimulus based on his 2018 tax return, including the child, you may still claim your daughter on your 2019 tax return.
*If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
A common error is when unmarried parents live together, If you and the other parent live together, only one of you can claim the child for any tax benefit. The interview is confusing (it's designed for divorced parents, who are allowed to split the child). The second parent should not enter the child, at all.
