Is it correct to say "the 'following' customer"?

It happens all the time. You are in line at the grocery store, Starbucks or anywhere cashiers are employed. Having finished a transaction, one will cheerily offer to help "the following customer."

I'm pretty sure that "the next customer" is the correct usage here, unless they call that following customer by name. Am I right?

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.

There's nothing wrong with it. In this case it's being used in the same context as "the following day" which is commonly interchanged with "the next day".

People line up and follow each other just the same as the days in a week. It makes sense to me!

According to OneLook it means:

adj. going or proceeding or coming after in the same direction, e.g. "The crowd of following cars made the occasion seem like a parade"

So although it is not an incorrect word, perhaps it grates on one's nerves since it connotes the sense of people as objects flowing mindlessly toward the cashier, as if you have no choice but "to follow".

It reminds me of a scene in a movie once with a tour guide directing a group of tourists to follow her, saying: "We're walking... we're walking... we're stopping..." The language it correct, but it's funny because it makes the relationship between the tour guide and the group seem so mechanical and impersonal.