This is possibly an unintended word play.

None of the other examples that you mention, Ents or eagles, give birth. The dwarves were not born, Though later we assume they would give birth.

Elves do give birth and so were the first born.

Is possibly a word play?

None of the other examples that you mention, Ents or eagles, give birth. Ents may germinate, eagles hatch from eggs. The initial Dwarves were created and were asleep. They did not reproduce until after they were re awoken

Elves do give birth and so were possibly the 'first born' in a literal sense.