The simple answer to your question is that Akeky Akekyavich becomes something other than human at the end of the story, when the author describes his actions after his death.
A rumour suddenly spread through St. Petersburg that a dead man had taken to appearing on the Kalinkin Bridge and its vicinity at night...seeking a stolen cloak, and that, under the pretext of its being the stolen cloak, he dragged, without regard to rank or calling, every one’s cloak from his shoulders...
The deeper answer is that Akeky Akekyavich is dehumanized right from the beginning of the story. He is ridiculed by his colleagues, his job consists of meaningless drudgery, and when he atttempts to recover his stolen coat, he is ignored and abused by government officials.
The only time that his life seems to take on any meaning is during the months that he is saving money to pay for his overcoat.
his existence seemed to become, in some way, fuller, as if he were married, or as if some other man lived in him, as if, in fact, he were not alone, and some pleasant friend had consented to travel along life’s path with him, the friend being no other than the cloak.