Sone of that was right, some was wrong. Using 1934 as an example, the Lasalle, Olds and Buick shared a basic chassis. The Buick engine was overhead valve 8, the Olds was a flathead 8 built by Olds and the LaSalle was the Olds engine built in the LaSalle plant.
Well into the 70s, each nameplate in GM and to a lessor extent, Ford, Mercury and Lincoln each had their own engines. Pontiac cars had Pontiac engines. Cadillacs had Caddy engines. In the 50s and 60s, GM didn’t even share rear axles. Transmissions also had significant variations.
By the 80s, more parts were common but the “tuning” of various parts were specific to the make. Shocks, springs, stabilizer bars, stereos, carpet, headliner materials, seats, exhaust sound, anything that could differentiate an Olds from a Buick or Chevy… or a Ford from a Mercury.
The 80s was when real badge engineering happened in the US car market. The Brits had been doing it since the early 60s as their car companies struggled and consolidated.