By that token, if KinGin were both in it, why would it only be named after 1 of them? It's no issue saying KinGin Kyoda", so "KinGin Butai" wouldn't be hard either. KinGin were a team, and treated one another as equals, even though Kin seemed to be stronger. So, if they had a team, leadership would be shared among them. They don't call each other "The Kinkaku Brothers."
I think people are making it more complicated than it should be. The verge of death. Kishi drew the Kage with blood splattered around them. KinGin bloodied them up when there, and they were left in critical condition. It was a physical beat down. Sealing doesn't leave blood behind-we saw that with Samui, Atsui, and Ginkaku. The Kage didn't get sealed.
I don't think it's as complex as some fans seem determined to make it. Databook states KinGin faught the Kage with those weapons, and Kyuubi chakra. But nothing was said of the Kinkaku Unit when mentioning the brothers. The Kinkaku Unit was mentioned on Tobirama's page, but didn't mention KinGin specifically in that confrontation. Kishi seems to keep them separate. Only thing shared is a name. It was a simple battle, where Tobirama and the Riakage lost. But Tobirama at least was able to get medical attention, and recover from his injuries, until his official death in the forest.
But if we have to get complex, I could say Kishi just used the name for the force as a reference of Journey to the West. But it was a good bit before the war arc, where he probably hadn't thought to include the actual Gold Horn and Silver Horn, from Journey to the West, along with their weapons. KinGin could have been a thought he had later, and thus we have this case with a character sharing a name with something else, even if they may not be affiliated, in the sense he was there. Sometimes authors don't plan absolutely everything out in advance. And I think Kishi admitted to that.
Just seems simple that it was 2 events.