Most supermarket chains sell oil from recognisable brands (Castrol, Shell, Mobil, etc). Some also sell their own brand. Most offers are on the big brand ones, so that is what I normally go with. It is often cheaper than the own-brand oil when it's a 2for1 or multibuy.
Not that I wouldn't use, for example, Asda-branded oil if it was available and the best price. It will meet the spec on the can. It wasn't made by Asda. They are not a petrochemical company. They are good at negotiating supply contracts, and pretty efficient at logistics. They would not risk selling oil with their brand on, if it wasn't completely satisfactory. The downside would be too great. I might have reservations if I was servicing a new hypercar, but chances are the supermarket wouldn't stock suitable oils for them anyway. Supermarkets are, after all, trying to sell lots of stuff, so they stock stuff which they can sell a lot of.
In the UK we do still seem to have some hangups about car-related things from supermarkets. In France it is quite normal to get your oil, filters, tyres, bulbs, brake pads, etc. from a hypermarket (I use the local LeClerc when I am there). Same in parts of Italy. Not had to service a car in other countries yet, but wouldn't be surprised if it was similar in most European countries.
Perhaps we are different here after all. But not better.