Factories that make the same product for different brands and different supermarkets use different recipes. This applies across the board - in the past 10 years I have worked with...
The company who make M&S shortbread (and also cheaper supermarket shortbread) and many branded biscuits - different amounts of butter used in the better quality ones is the main difference.
2 companies who make ham for all the supermarkets - different amounts of water, and different processes for reconstruction of the meat depending on who their end client is.
Several companies making ready meals for everyone from Aldi to M&S (same manufacturer often making for both ends of the premiumness spectrum). They make what is nominally the same product (Chicken Tikka Masala, for example), but different recipes - different spices, levels of cream, amount of chicken, sizes of chicken pieces, quality of chicken, etc etc.
A cereal manufacturer - they make branded and OL... to different recipes.
2 big dairy companies - both making branded yogurts and desserts and selling OL yogurts... again, different recipes.
When a supermaket enters into a contract with a manufacturer to produce a product, the recipe for that product is developed specifically for that supermarket, and significant testing and tweaking takes place (to make sure it's the right product for their customers and can be sold from manufacturer to supermarket at a price where the mftr can make a profit, and from supermarket to customer at a price where they can make a profit). The manufacturer can't change the recipe without going through a process with the supermarket.
There might be similarities between recipes for different products, but they are exclusive to the retailer they're being sold to (or the brand, in the case of branded goods).