Can my fiance claim me and my daughter as dependants when filing his upcoming 2019 taxes?

[Critter]

Maybe IF you qualify as a "qualifying relative" per the IRS regulations :

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 are 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.
• They made less than $4,150 in 2018 ($4,250 in 2019).
• You provided more than half of their financial support.

[Hal_Al]

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit.  So, your child may qualify as a  dependent, but because he is not related, she cannot be a qualifying child for the earned income credit, child tax credit or Head of Household filing status. The Other dependent (qualifying relative) credit is worth (up to) $500 per dependent and is non-refundable.  That is, it can only be used to reduce an actual tax liability.

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year (not even one night at the non-custodial parent’s home).
2. His/her gross taxable income for the year must be less than $4250 
3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer

He must have a US social security number or tax identification number (TIN)

In addition to the above requirements, to claim your boy/girlfriend/fiance's children, they must meet all of the above requirements and:
--- your fiance must not be required to file a return,
--- he/she does not file a return claiming the children

[Bees]

Is your daughter his biological child?  If so he will be able to claim child tax credit.