Are Shabbat robes Tzanua? Can they be worn outside the home? Is it Tzanua if they resemble garments worn by Muslim women, such as abayahs?

Tzniut is a question of Halacha and if the garment fits the Halachic requirements – it is Tzanua. Nonetheless Tzniut is a question of geography, to say - another factor for Tzniut which is does it attract special attention, and if it is something that is not worn in that specific place and time it might not be Tzanua to ware as you stand out and attract attention. I'm pretty sure these robes meet all Halachic measures of Tzniut.