It is indeed true that the big oil companies supply petrol to the UK supermarkets. Petrol sold in the UK needs to meet British Standard 4040 (The Motor Fuel (Composition and Content) Regulations 1994
) but the oil companies have more specific standards for their own-brand petrol.
For instance, Shell has one of the highest detergent fuels, which they claim keeps your fuel system and valve-gear cleaner, and indeed many motorists seem to agree.
'Off-specification' petrol that still meets BS4040 (or can be blended with other batches to make it comply) is sold on the oil spot-markets at a reduced price - and much of it is bought by the big supermarkets.
What all this means is that most oil company forecourts sell very predictable fuel that'll be the same every time, but supermarket fuel is less predictable: it could be just as good as the branded stuff, or it might have just
scraped over the BS4040 spec.
Very unlikely that any of these fuels will damage your engine (because they all have to meet the minimum spec that was agreed with the motor manufacturers), but if you want consistent performance then Shell is probably one the best at the moment...