Specifically, I was reading the Sengoku RPG book — which I know not to be a historical source as Academia is concerned, but is still pretty well written and well-informed.
I found in a website that a "mou?" was a half-apron, but the question remains.
I believe it would hold the bottom half, not the upper one, due to its color. However, how it was used and the details of the appearance still eludes me.
Its mo, not mō: the kanji character you are looking for is 裳. (It is a pain to google, since the character also appears in combinations, with 衣裳 being what complicates things.)
Notably, in Heian period court society, the traditional rite of passage for girls transitioning to adulthood was called mogi 裳着—wearing a mo—so the dress signifies being an adult woman, and thus socially perceived as being marriable. It is no surprise that often, this was performed only shortly before a marriage. With the passing of time, it also increasingly was abbreviated or omitted, and, as far as I know, not performed within lower social ranks.
Japan’s Cultural Heritage Online resource site has examples of the garment here and here , although there is no information given on the specifics of these pieces (I suppose it is safe to say we're likely talking Edo-period here). Likewise, Wikipedia would give you this one here (Edo-period), apparently displayed in Tokyo National Museum (although I couldn’t find it in the National Museum’s digital resources itself).
As you can see, these are not red (well, okay the latter kind of is). I'm indeed not sure why the game-designers speak of red specifically (neither are history of clothes nor the Sengoku period my field, tbh), but there used to be some regulations pertaining to the colors to be used for in place, esp. for people of lower status, way back in the Heian and Kamakura period. Still, that would never generalize into just one color for everyone.
