What you are describing as a "mushroom" flavor is most likely umami, loosely translated as "savory". It is a flavor that comes from glutamates, which are found in foods like meat, mushrooms, tomato paste, and soy sauce. MSG (Monosodium Glutamate) is a refined/artificial version that can add an umami taste to food, which is often perceived as a "meaty" taste, but since mushrooms contain a lot of glutamates, what you are perceiving as a "mushroom" taste is quite likely the same.
Additionally, ingredients like yeast extract contain glutamates as well, and added flavors, both natural and artificial, contribute to the taste of manufactured soup.
Campbell's soup contains the following ingredients that give a savory/umami flavor, as do mushrooms:
Soy Protein Isolate, Monosodium Glutamate, Flavour, Yeast Extract
Batchellors soup contains the following:
Flavour Enhancers (Monosodium Glutamate, Disodium 5' - Ribonucleotides), Yeast Extract, Flavouring
and Heinz soup contains the following:
Yeast Extracts, Flavouring
so it appears that all the soups you mentioned are using ingredients not commonly found in home kitchens to amp up flavor.
You can try adding umami flavor to your soup by using lots of umami ingredients, such as tomato paste (also found in Campbell's soup).
Otherwise, the only way to acheive such flavors might be through MSG and similar flavor enhancers.