Bodies have genders (physical anatomy and genetics) and the brain has a gender (gender identity), but do the spirits/souls or our heavenly bodies have genders as well?
- Do we retain our gender in Heaven? at Christianity
The Zohar writes that the Patriarch Yitzchok had the soul of a female.
Additionally, the Arizal explained that Michal wore tefillin because she had a male soul.
So it seems that souls do have genders, both male and female.
No. A soul is a spiritual entity and has no gender. However according to Kabbalah there are two worlds where souls can come from. A "Masculine" world and a "Feminine" world. The Lubavitcher Rebbe Z"L explains that that
So when the Kabbala uses those terms it does so as a metaphor to explain a concept not to indicate gender.
Note when explaining Michal the daughter of Shaul the holy Ari says that she had a
loosely translated as "She had a soul from the world of the masculine (side)" Not that her soul was of the male gender.
I encourage you to read the whole article from the Chabad website - Female and Male Souls
the shemonei shearim shaar gilgulim (forget exact place) says that if a male soul is reincarnated in a female body, as sometimes happens, then that female cannot have a male child - only female children, as the verse says in parsha tazria "when a woman conceives and bears a male child", and that woman is really a man in a female body.
I think he says there that male soul are from adam while female souls are from chava.