"The Chaser" is told from a third-person objective point of view. The narrator is not omniscient but only reports what can be seen and heard in the single setting, a little shop in a big modern city. There are only two characters in the story, a young man who is madly in love and an old man who sells what would appear to be magic potions, although he never claims they are magical; they could be secret mixtures of his own creation, or they could be nothing but colored water. The story reads like a play. Most of the information is conveyed in dialogue. The author would have no reason for telling the story through the point of view of Alan Austen or that of the anonymous old man. Austen's psychological perspective is conveyed through what he says and does, and the same is even more the case with the old man.
The focus of the story is on the old man because he is the vendor of the mysterious potions and has had so much more worldly experience than his customer. We realize that he is amused by Austen because he has seen hundreds of others just like him--young men who are madly in love and think it will last forever. We can imagine everything the old shopkeeper is thinking without having to be inside his point of view. If we were inside his point of view, all we would see would be a nervous young customer. If we were inside Austen's point of view, all we would see would be this strange shop and the strange little old shopkeeper. Without being confined to either character's point of view, it is as if we are spectators at a stage play and are watching a drama being acted out.
"The Chaser" is an example of metafiction. See the e-notes reference links below. A good contemporary writer of metafiction is Woody Allen, also a frequent contributor to the New Yorker magazine like John Collier before him. Please refer to the reference link on Woody Allen below. A common feature of metafiction is that it is not to be taken too seriously. If Collier had written "The Chaser" from the point of view of either of his two characters, it might have made the story seem more serious, and that was not his intention. He does not want to explore Austen's feelings of love too deeply, and he does not want to take the old man's business too seriously either.
The story is told in the omniscient third-person point of view which is the most common point of view used in story-telling. It is really structured just like any story about a person who comes into a shop to buy something--except that the buyer is after an unusual item and the seller has that item in stock. There is something very mundane about the transaction--just one of thousands of sales that take place in a big city. However, the narrative emphasizes the perspective of the old man, since it is his shop and since he obviously understands the young man's problems much better than the young man understands them himself. When Alan Austen leaves with his magic love potion, the old man knows he will be seeing him again. Instead of saying goodbye, he says, "Au revoir," which is about the same as "I'll be seeing you."