"egg-having partner" is an extreme far left term that tries to work around the hole they dug for themselves involving transgender ideology, that the vast vast majority of the world does not care for. The non-political term is women or wife or female or anything else along those lines.

There are women, wives, and females who do not have eggs though.

No one dissects children to inspect whether they have egg cells when they're born. Gender is assigned solely based on the doctor's interpretation of their genitalia appearance.

I honestly don't care what words people use though, as long as I can understand the words? Egg-having partner would fit that description to a pedantic degree.

y’all are the ones injecting politics into it, ya weirdos.

how would you specify which woman you mean in a same sex couple?

i am personally not in need of this term but think its an especially good fit in this specific context.

context being the defining factor, i would usually not appreciate explicitly reducing a person to their reproductive organs like this but in a discussion about reproduction it actually removes ideology from my point of view.

Not that term for sure. Both of those women have eggs. The descriptor you're looking for seems to be "pregnant".