WMG / Final Fight

Poison is actually a case of complete androgen insensitivity syndrome
Complete androgen insensitivity syndrome is an actual condition in which a person will have male chromosomes, but develop physically female. Poison is always depicted with a female body, including very large breasts, and in recent games is voice acted by a woman. Despite Capcom's inability or unwillingness to outright say what Poison is, this would make more sense than Poison being a cross-dressing man or undergoing Easy Sex Change surgery- and being realistic, a low-level street punk with no money and a probable criminal history probably isn't paying for any extensive surgery. CAIS would likely better explain the gender confusion without breaking current canon or going against her character artwork.

Poison bought a cloned DNA-altered body
On the other hand, this is the same canon as Street Fighter and eventually Megaman so there is plenty of sci-fi to work with in regards to Poison's transgender state. We know cloning tech exists as M. Bison has used it to cheat death multiple times(I will safely ignore the fact you can fight multiple copies of the same enemy in Final Fight, even multiples on screen at the same time, as there is no in-game or canon acknowledgement of this to indicate they are cloning gang members) and Bison created female clones of himself using genetic alteration(Cammy, etc). It's reasonable that, even on the black market, someone with enough cash to get access to Shadowloo tech to buy a new cloned body and have it altered to their preferences. Any references to surgery could suggest the method of transferring mind or brain as opposed to standard real-world cosmetic alteration.

Any canon references to a "pre-op" state could indicate Poison had some degree of standard surgery prior to the new body, possibly stopping just before genital reconstruction.