As you will know, but as those living outside the UK may not, the names the supermarkets themselves use are Sainsbury’s, Morrisons, Tesco, Asda, Co-op and SPAR. The first two are named after their founders, so the terminal ‘s’ indicates that it is, or once indicated that it was, Sainsbury’s shop and Morrison’s shop (and let’s leave aside for the moment why there’s an apostrophe in one and not in the other). Co-op is short for The Co-operative, so there’s no semantic reason why it should be known as Co-op’s. Similarly, SPAR, I believe, is an acronym, so equally SPAR's would make no sense. That leaves Tesco and Asda. If the first is sometimes called Tesco’s, that may well be by analogy with Sainsbury’s and Morrisons. That is not the oddity. The oddity is that Asda is not known as Asda’s. At least, it’s not as far as I know. Maybe in some quarters it is. If it’s not, perhaps it’s because it was a relatively late arrival and the name Asda was thought somehow to be inviolable.
You may know that the UK bookstore chain is changing its name from Waterstone’s to Waterstones. This has provoked outrage among the usual suspects. However, a company can choose to punctuate its trade name how it likes without regard to what is done elsewhere. Hence Sainsbury’s but Morrisons. There are three sensible and well-informed posts on the topic here, here and here.