It does happen that smaller car makers will use parts from other makers in their cars because it is too expensive to design and produce all of the parts themselves. Switches, tail lights, etc are often done this way.

It is not uncommon for parts that are made by third parties will be used in cars by different makers. Aisin makes transmissions used in many brands. This can backfire when the third party maker has a production problem which affects a wide range of cars in multiple makers. The Takata air bags were used in many brands of cars until they were found to blast shrapnel into the faces of occupants. Takata has been struggling for years to produce enough replacement parts to fulfill all of the recalls.