To my ears, go to the Asda sounds absolutely fine. But that might be because I've never heard of Asda before now. So instead, I tried dropping in the name of a grocery store I'd go to — say, go to the Publix (a chain of supermarkets in the southern United States). And it does sound strange.
But that might be because it's out of context. What if there's a Tesco a block away from the Asda and both you and your friend know this?
— Let's stop and get some digestives at Asda.
— We can't. They're closed.
— Well, all right. Let's go to the Tesco.
If the last speaker intends for the the other to pick up on which Tesco is being referred to — and if this debate is so long-standing, perhaps your friend does have this expectation :) — perhaps this usage can be justified.
To use the technical term, a definite article triggers a presupposition, another statement that must also be true in order for the original statement to make sense. In general, a phrase like the something-or-other presupposes 1) that something-or-other exists and 2) that there's only one something-or-other. At least that's how it's usually summarized. In reality, the second part often works like so: that there's only one relevant something-or-other (e.g., go to the bank).
If that's what your friend's intention is, he may be justified in his usage. If you don't feel he could rightly assume you'd know which one, perhaps he wouldn't be justified in his usage. How's that for a copout? :)
Beyond that, I disagree with Tristan's comment above: This question is not related to the one on go to hospital. Those phrases are what Jespersen (and for all I know others) called unproductive constructions. In other words, go to hospital is a readymade construction. You can say go to school or (if you're British) go to hospital, but you can't say go to store or go to bank. (The example I remember from Jespersen is his contrasting Long live the king! with Soon come the train!) You can say go to the prison, but it doesn't mean go to prison. Ultimately, a phrase like go to the store or go to Wal-Mart is not taken off the shelf fully formed; rather, the speaker has to assemble it specifically for what he or she means.
(By the way, did you notice what I did in the second sentence of the second paragraph? Just curious.)
Best Answer
It’s a participle being used as an adjective (like “the walking dead” or “the setting sun”). It’s a little unusual, in that the phrase “next customer” is the common idiom in English, but I can’t see what could possibly be wrong about it.