Which brands are the same?
Trane isn't part of Bryant & Carrier. It's owned by the Irish company Ingersoll Rand along with American Standard.
As for the same, none of the below are 100% the same as their sisters, though some come pretty close any more.
Goodman has Goodman & Amana
Carrier has Carrier & Bryant
Lennox International has Lennox & Allied (which is Ducane, Armstrong etc)
Goodman and Carrier both have app. 25% of U.S. sales. The other 50% by others.
If I remember correctly, there are only seven major manufacturers of HVAC equipment, however this equipment is sold under about 50 different brands. United Technology has about 10 brands, Goodman about 4 brands, American Standard (Trane also), Lennox about 4 brands, Rhemm/Paloma two brands, York about 3 or 4 brands, and Nortek about 7 or 8 brands. Anyway the actual companies are far fewer than brands they make. And the largest selling single brand? Goodman (now you know why some contractors hate Goodman, especially if they lost sales to them)!
P.S. I agree with Cosmic on the ranking too.