So, there's this utterly meaningless detail I got it in my head to check on, and for all the digging I did it produced nothing... But i share this nothing now, with you.
There's this particular cloak Kuukaku throws on when she goes to recruit Jidanbou to fight the other Gate Guardians so she can go help Yoruichi try to stop Aizen. And it is basically the same design as the cloak Byakuya is wearing in Renji's flashback when he finds out Rukia's been adopted into the Kuchiki family. Presumably it's some kind of royal family attire. It's possible Yoruichi's family head wore something similar with their family crest on the back; We only barely see what they're wearing but it does seem to have the same high collar with a narrow trim. (I was wondering if the Shihouin cloak Ichigo wore for all of half a second was of any kind of similar design, so went back to add it to this collage too, but it's not really similar. Oh well, that might have been kind of a cool detail if it were)
Anyway, the next time we see Kuukaku Kubo forgot to draw the binding rope with the tassels draped across the chest... And even in the final arc where she's shown wearing it again, he didn't remember that detail.
So I was wondering if Byakuya put his back on in the final arc, but the design isn't quite the same, although it shares some features... But sort of like Kuukaku's it also fluctuates in design from one appearance to the next...
It starts with two lines of trim along, making it very distinct from the family cloak, and it has a sharper edge than the rounded collar, and a more ornate set of tassels.
But then when Byakuya unfastens it, the broad trim that clearly wrapped over his shoulder before gets changed into a lapel, the collar loses its double trim, and gains a new tassel holding the breast together, even though that wasn't there before. Also the ornate tassel on the collar gets simplified.
Then he loses the lapels as battle damage in the middle of the arc... (but gets them back when fighting Gerard after having been offscreen for quite a while.)
Ironically in this process of constant redesign/misdrawing the robe it actually wound up looking just like Kuukaku's cloak after all, including the edges getting rounded out, and the absence of the cord across the chest
and unrelated to any of that, his epilogue coat is yet another clean white design with a thing gold trim, but the collar is folded down, and doesn't appear to have any tassels. It appears to be a highly specialized captain's coat? We never see the back to check for a squad's numbered insignia so it's impossible to tell.
...i forgot he was in the Fullbringer arc too. and this just muddles things furthers... this one definitely is his captain's robe but looks even more like the family cloak in design. This doesn't clear up any on whether this is all one robe that Kubo keeps forgetting/changing the design of, or if they'll all different but very similar.
Thr Fullbring and epilogue ones are both specifically captains robes, sleeveless with holes for the arms. But Kuukaku's and the one Byakuya gets from Shutara are cloaks that go over the rest of their clothes.
Anyway like I said, there's not actually a take away from this. plain white cloaks with gold trim look similar, go figure ¯\_ (ツ)_/¯