Female = noun
The way (someone gives birth) = noun
To give birth = verb. An action.
Feminine = adjective. A word that describes a noun.
For example:
- The woman gave birth
- The feminine woman gave birth
- The woman gave birth in a feminine way
What makes the woman in this sentence feminine?
The only possible answer is that the observer who wrote the sentence has decided that whatever she was doing prior to or during the birth was stereotypically female enough that it warranted the adjective being added. If she was simply giving birth - an action that only a woman can do - the word "feminine" would be superfluous.
Perhaps prior to and during the early stages of labour she was wearing a selection of floral dresses, baking cakes and softly smiling at everyone while whispering gently about soft, fluffy kittens... and then when things got going she did a series of beautifully delivered pirouettes in the style of a jewellery box ballerina until the baby was born.