It is possible to claim a non-relative on a tax return if they meet ALL the requirements under the Qualifying Relative rules.
To be a Qualifying Relative -
1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross income for the year must be less than $4,200 (social security does not count) in 2019
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.
6. The person must not file a joint return with their spouse.
If all of you have lived together for the entire year of 2019, AND you are not claiming your own children (did you not have any income?) then BF *might* be able to claim the children as "qualifying relatives" even though they are not related to him. He cannot use them to file as HOH or to get the child tax credit or earned income credit. He can get the $500 credit for other dependents for them.
CREDIT FOR OTHER DEPENDENTS
IRS interview to help determine who can be claimed: