Imitating the flavour of canned cream of mushroom soup

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.

A number of years ago I had access to literally a ton¹ of fresh mushrooms (the common grocery store type) and I set about trying to duplicate Campbell's cream of mushroom soup.

I landed on a very simple recipe: basic milk-butter-flour bechamel, using about one cup flour per gallon of milk. Melt the butter and sauté the finely-chopped mushrooms until they're dark and most of the water has cooked out of them. Add some celery and onion, then sauté until the celery is soft. Add flour for the roux, cook a minute or two, then add the milk gradually and let it thicken. I prefer to go at it with a potato masher for a while at this point; optional. Salt and pepper to taste.

I've also made this with home-canned mushrooms (from the same event :- ) rather than fresh and I couldn't tell the difference in the final product.

¹  I volunteer at a food pantry. A trucker had a delivery of two pallets of fresh mushrooms refused by a local grocery store (for a complex but understandable reason). There were other goods on the truck behind the mushrooms and so he donated them to us in order to be able to continue his route. Fresh mushrooms do not hold well... We called every pantry in the region trying to get them moved before they turned to slime. Volunteers were invited to take as many as they wanted. I fired up my 20-quart Presto canner :- ) and we still ended up dumpstering a lot of them.

Well, this sounds like imitating the imitation. Are you guys cooking industrial made "mushroom" sauces or soups only? Why not buy dried mushrooms (boletus, chanterelles and others, that are "the real stuff"? Depending on where you live and what you can buy in supermarkets or green markets, go for the real stuff. Even dried ones. You can soak those dried mushrooms in tepid water for a few minutes, up to 20, (depending on the sort and size or the mushrooms) then use them as fresh ones. You'll get everything, flavour and texture, but especially FLAVOUR. After you've soaked dried mushrooms, treat and cook them as you would the fresh ones.