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

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.

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

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