Symbols on Banners and Sails

I'm just getting started with my Dark Eldar army, and I was wondering a bit about the symbols on the sails and banners.

I noticed that they are either the kabal symbol, coven symbol, or wych cult symbol. I would like to stick to one symbol, but I don't know how the fluff works exactly.

In the fluff, do the kabal, coven and cult work together under 1 flag/symbol? I like the obsidian rose for my kabal, but I'm not quite sure how the rest of it would go. Does anyone paint different symbols for different units?

Id like to nail down my scheme before I get started. I really like the Obsidian Rose paint scheme, but not too keen on the symbol when compared to other stuff

No, they have their own.
Each Kabal has a symbol, as does each Cult and Coven.
THink of it like an allied army. The British will go to war under their own flag, and alongside them the Americans with their own flag, and maybe the French with their own.
Kabals, Cults and Covens going to war together is a temporary friendship or alliance, or a bartering of services in exchange for other things. But though they act together, they are not entirely united.

In terms of using the Obsidian Rose scheme, theres nothing stopping you from using a very, very similar paint scheme, but using a different image and giving them a different name.

Sharing heraldic elements durring campaigns is mostly something Imperial forces tend to do, but Eldar aren't above it either. For example, Alaitoc's many Ranger and Exodite allies usually wear the Craftworld's blue, yellow and white. And Lugganath's troops wear the same colors as the Sunstorm Brotherhood of Corsairs and even diamond patterns based on Harlequin holofields.
As for Dark Eldar: Wych Cults often adopt the color sheme of the Kabals sponsoring them, as is the case with the Black Heart and the Cult of Strife, the Poisoned Tongue and the Seventh Woe, or the Flayed Skull and the Red Grief. Mercenaries like the Scourges, Incubi and Hellions do this if they are permanently hired or otherwise have special ties to a certain organization. I don't think they would be above displaying their partner Kabal's symbol alongside their own either.
Haemonculus Covens and Mandrakes always stick with only their own colors and symbols (I don't think Mandrakes actually have any symbols, but very little is known about their social structures) though, and usually have connections to multiple Kabals and Cults without being really tied to any.

In my backstory my twin Archons Diaval and Sinyxl run everything, but their Kabal and Wych cult have their own logos so I created a third organisation run by the twins which emcompasses everything they do outside of the Kabal and Wych cult, an umbrella organisation so to speak called the Dark Gemini. Their slave trades, scourges etc all fall under this category. I haven't come up with a logo yet but perhaps its something to think about.

Of course if your Archon is particularly pedantic about their logo they may insist that all elements that accompany them on a Realspace Raid do so in the colours an under the symbol of their Kabal.

The sweeteners that get thrown in to entice the Haemonculus covens might be well worth it for the extra brand exposure Got to look after that branding, there's no point laying waste to a sector if no-one knows who did it!

Maybe your Archon, to sweeten the deal, provides all vehicles used in the raid. This would reduce the costs for the Succubi and Haemonculi who are participating in the raids meaning more profit in the end.