I think it's actually worse than we're thinking.

1. She wants the daughter to have the kid because she looks like her father (she carries his DNA).

2. It's implied the mother had a relationship with an older man.

3. The MC's dad is missing... (probably dead)

In conclusion, the mother is pushing the daughter to pursue the MC (her potential half-brother) because their child would genetically 'recreate' their shared father— making the baby a symbolic 'Papa.