I think, it’s probably more of a rights to the name timeline issue than an ownership overlap.
It’s confusing, all these years later, but when Frederic Robie sold the assets from the Excelsior Motor Manufacturing and Supply Company, he sold off separate assets to separately interested parties.
Arnold Schwinn and Company only bought the rights and tooling that pertained to the motorcycle business,
They didn’t need or want the bicycle rights and tooling, so that was sold to what would become, the Excelsior Cycle Company.
They continued to operate in Chicago for a few years, before moving to Michigan City Indiana, and a close relationship had been established with the retail outfit, Chicago Cycle Supply.
When Robie sold the company, all three entities bought the bits and pieces.
So, it’s possible, that the name Triumph, was something that Chicago Cycle Supply picked up in the process.
They never actually manufactured bicycles, so the bike it’s self, would’ve been made by one of their suppliers, and they would’ve just put that badge on it, for sale out of their supply house.
Because they all came from the same origin, there were similarities between the various outlets and manufacturers who sold the wares afterward.
It is odd, that there weren’t any legal disputes between the joint use of the Excelsior brand, but it may have been that both companies bought the rights to the name, and there wasn’t ever any contractual difference established between the Motorcycles and the Bicycles, so they both had a legal right to use the name, and that’s just the way it is?