Yes and no.
You can review the rules in publication 501.
https://www.irs.gov/pub/irs-pdf/p501.pdf
Your DP can only be your dependent if they live in your home the entire year (as a member of your household), and you provide more than half their support, and they have less than $4700 of taxable income.
The rules for claiming their child are the same for residency, income and support. In addition, if another taxpayer qualifies to claim them, you can't, even if the taxpayer who could claim them agrees not to claim them. In the case you describe, the biological parent could claim them, but the biological parent is not a taxpayer as long as their income is less than $13,850 and they either file no tax return, or they only file to claim a refund of withholding and claim no other dependents, credits or deductions. Since you can't claim the DP unless their income is less than $4700, the DP is probably not a "taxpayer", and that will allow you to claim their child.
However, an unrelated child only qualifies as an "other dependent" for a $500 credit, not the $2000 child tax credit. And neither the DP nor their child is a "qualifying person" for head of household status, so you have to file as single.
