Generally, short lines are/were separate companies. However, it wasn't uncommon at that time for a large railroad to have bought a controlling interest in a smaller connecting railroad at some point in the past, but have the smaller railroad essentially continue to operate as a separate entity (it's own name, paint scheme, officers, etc.)
Also, many large cities had terminal railroads, where several large railroads would create a smaller railroad that would move interchange freight from one railroad to the other. For example, Lake Superior Terminal and Transfer was owned by the six large railroads serving Superior, Wisconsin. The problem was that if say the C&NW wanted to move a cut of interchange cars to the NP's yard, the C&NW would have to move the cars to the NP and return light (just the engine and caboose). If NP had cars for the C&NW, an NP crew would have to run the cars to the C&NW and then return light. A terminal railroad like LST&T could take the C&NW cars to the NP, then pick up the NP cars and take them to the C&NW in one round trip.