"Sainsbury's" supermarket used to be "J. Sainsbury's" named after its founder John Sainsbury, &c. "Morrisons" was "Wm Morrison" founded by William Morrison. So when you refer to a shop you say Sainbury's as in [Mr.] Sainsbury's shop, or "Morrison's" as in Mr. Morrison's shop.

Then this becomes so ingrained it gets misapplied sometimes. I don't think I'd ever say Asda's though. But I would say Tesco's, even though Tesco is the initials of three people.

Surely this is the same worldwide?

Completely normal to say "Tesco's", "Aldi's" etc.