It is easy to treat Achilles as literature and forget that he received worship. Hero cult in the Greek world was a real institution, distinct from the worship of the gods: it was tied to a tomb, the offerings went down into the earth rather than up in smoke, and the hero was expected to act locally, for the community that held his bones.
Achilles had such cults in several places, and they were not minor. There was a tumulus in the Troad identified as his tomb and visited by travellers, including, famously, by conquerors who wished to associate themselves with him. Around the Black Sea he was worshipped widely, and the island the Greeks called Leuke, the White Island, off the mouth of the Danube, was held to be where he lived on after death with a chosen company, with a temple and a body of sailors' reports of hearing him.
The relation between the cult and the poem is not simple, and the cult may well be older. What the two together show is a figure occupying a position the modern categories handle badly: not a god, not a fiction, but a dead man whose power was thought to be locally available and worth the cost of sacrifice. The atlas holds the Greek heroes in the same table as saints and sages for that reason, since the practice around them was closer to that than to anything literary.