Wait hold up. Why does wikipedia say that the toga is a Greek garment? The toga was characteristically Roman, to the point that the Romans were called togati, "the togaed ones." it was a characteristically ethnic garment not shared by the Greeks. The toga probably originated from the Etruscans, who are depicted in frescoes as wearing what is clearly a short form of the garment. The Greek ἱμάτιον was a broadly similar garment, in that it was a large, (usually) woolen cloak, and Greek sometimes used it as a synonym for toga. But the ἱμάτιον was a cloak, an outer garment for protecting against the weather--by contrast the toga was a garment of public significance, the distinctive ethnic garb of the Roman citizen in the public eye. Closer to the Greek ἱμάτιον was the Roman pallium. Both were rectangular (where the toga was semicircular) woolen cloaks worn purely for comfort in the weather and lacking the distinctive decorative elements that appeared on togae to distinguish between ages and status--indeed, the tradition was that the pallium was a Greek garment introduced after the toga and that it was originally only worn by philhellenes. That they were further considered different garments is reinforced by the fact that increasingly in the later Principate the Greek cloak began to replace the toga, which was larger, heavier, and hotter than the comfortable pallium.

Certainly initially the toga was worn by both sexes, originally as the only garment (never a norm with any Greek cloak). At this early stage in the city's development the city was quite rustic and unsophisticated. Already by the beginning of the Republic traditionally women had been introduced to the stola, around the same time that the tunic entered the city's dress. The introduction of the tunic meant that the toga was no longer the only garment worn, and the toga quickly was transferred to an official public garment worn outside for public functions. It should perhaps not be considered especially odd then that as male dress transformed female dress did the same. The introduction of the stola, a more stylish, closer-fitting women's dress, rapidly took hold, and by the late Republic the toga's use among women had become restricted to prostitutes (I presume because of the public associations with the garment, although I've never bothered looking into it)