The robe is the visible sign of ordination, so the question of who may wear it is the question of who may be ordained, and in contemporary Buddhism that is not settled.
The Buddha ordained women. The bhikkhuni order existed and is described at length in the Vinaya. In the Theravada countries, however, the full ordination lineage for women lapsed, in Sri Lanka around the eleventh century, and it never established itself in Thailand or Burma at all. Because full ordination requires a quorum of already fully ordained bhikkhunis, a lapsed lineage cannot restart itself, and that is the crux.
The East Asian lineage, transmitted to China and thence to Korea, Taiwan and Vietnam, did not lapse. From 1996, ordinations in India and Sri Lanka drew on that lineage to restore the Theravada bhikkhuni order, and Sri Lanka now has a substantial bhikkhuni community.
The Thai and Burmese hierarchies do not accept it. Thailand's Supreme Sangha Council maintains a prohibition dating to 1928 on Thai monks ordaining women; Thai women who wish full ordination are ordained abroad and are not officially recognised on return. In Burma the position is similar. The consequence is visible in the colour of clothing: women practising seriously in those countries wear white as mae chi or pink as thilashin, holding eight or ten precepts rather than the full set, without the legal standing, the alms rights or the institutional support that go with the ochre robe.
Tibetan Buddhism sits in between. The bhikshuni lineage was never transmitted to Tibet, novice ordination for women is available, and the Dalai Lama has stated repeatedly that he supports full ordination and that the decision requires broad monastic agreement rather than his own ruling. Progress has been slow and is contested on procedural rather than doctrinal grounds.
So the robe, which in its origin was pieced together from cloth nobody wanted, is now the object at the centre of the most consequential status dispute in contemporary Buddhism.