Is there a name for the category containing the words trans and cis?

Cis and trans, standing alone, used as adjectives, in relation to gender, are gender identity terms. They are shortenings of 'cisgender' and 'transgender'. Cis and trans are used as prefixes in academic writing (mainly chemistry) to mean "on the same side" and "on the opposite side" respectively.

Cis
Trans

Wikipedia says "Gender identity is the personal sense of one's own gender. Gender identity can correlate with a person's assigned sex at birth or can differ from it." Whether you are cis or trans means whether they are or are not so correlated, respectively.

"Trans status" is used for the concept the OP seeks a name for. Unfortunately, "X's trans status" could be misunderstood as "the fact that X is trans" rather than "whether X is trans or not". (Much as "the film's length" could be taken as either "how much time the film lasts" or "the fact that the film lasts a long time".)

"Sexual Orientation, Gender and Trans Status Monitoring" -- Brighton & Hove LGBT Switchboard (Note Gender and Trans Status being listed as two different concepts.)

"Getting Equalities Monitoring Right" -- Scottish Trans and Stonewall Scotland. p.7 "If they dont' want to share their sexual orientation, gender identity or trans status with their employer, they don't have to."

I believe you're looking for gender identity

"My gender identity is cis." Or more commonly "I identify as cis."

Cis and trans being two, of many, ways that people identify their gender.

Transgender offering:

• noting or relating to a person whose gender identity does not correspond to that person’s biological sex assigned at birth

Cisgender offering:

• Also cis·gen·dered. noting or relating to a person whose gender identity corresponds with that person’s biological sex assigned at birth.