As a matter of fact, 3 ancient sources report this: Diodoros of Sicily, Curtius Rufus, and Justin. But the passages are all very similar, and probably go back to a single author whose work is now lost. Here's Diodoros 17.77.6-7:

[Alexander] added concubines to his retinue in the manner of Dareios, in number not less than the days of the year and outstanding in beauty as selected from all the women of Asia. Each night they paraded around the couch of the king so that he might select the one with whom he would sleep that night.

The anecdote is part of the mythologising, sensationalist tradition about Alexander the Great, which sprang up immediately after his death and has always been a rival to the more sober political-military accounts of his life. The former was more focused on gossip and spectacle, and stressed the supposedly outrageous decadence, but also the luxury and sensual allure of the Persian Empire. This fits into a long tradition of Greeks writing accounts of the Persians that we now recognise as "Orientalist" - relying on specific sets of stereotypes to draw imagined distinctions between East and West. In both Diodoros and Justin, the account of the 365 concubines follows directly after another story that was meant to illustrate the strange but sexy world in which Alexander found himself: the story of the Amazon queen Thalestris, which I wrote about here.

But there's more going on. In both sources the anecdote comes with a whole bunch of other things that Alexander started doing "in the manner of Dareios" - using Persians as servants and bodyguards, dressing like a Persian, equipping his cavalry in the Persian fashion, and so on. In other words, this isn't just a story about the enticing East. It describes a key stage in Alexander's supposed corruption by Eastern values and practices, which would ultimately cause his Macedonian army to lose faith in him and rebel against him. If having 365 concubines seems bizarre and excessive, it's because the story is meant to make Alexander's behaviour seem bizarre and excessive. A Macedonian ruler wasn't supposed to dress like that, nor to trust Asians with his life, and he certainly wasn't supposed to indulge in the exploitation of a vast harem. Diodoros actually notes Alexander's anxiety about this immediately after the story of the concubines:

Alexander, as a matter of fact, employed these customs rather sparingly and kept for the most part to his accustomed routine, not wishing to offend the Macedonians.

(There's a lot more that can be said about the strong links between oppressive tyranny and sexual excess in Greek and Roman thought, but we can leave that aside for now.)

So is this all just a story, told to make the East look like a place of despotic decadence? Maybe. We happen to have an independent source confirming that it was customary for Persian royalty to maintain large numbers of women for sexual exploitation. Plutarch records the report of Ktesias, a Greek doctor at the court of the Persian king Artaxerxes II (r. 404-358 BC), that besides his wife the king had 360 concubines at his court, "each one of superlative beauty" (Life of Artaxerxes 27.2). This could be just another example of the same Orientalising tradition, or it might indicate a real custom of granting the king a harem of as many women as there were days in the year.

If we assume the story about Artaxerxes II is real, then we may suspect that the story about Alexander is real as well. It's understood that Alexander's adoption of Persian ways and attempts to integrate the Macedonian and Persian elites through intermarriage were attempts to legitimise himself as ruler over an empire which rested on more than 200 years of royal ideology very different from that of his own country. He was an outsider; he needed to show respect to Persia or he would never be accepted as its king. It is possible that the acquisition of 360 concubines was another way in which he tried to signal that he was a legitimate king in the Achaemenid mold; it's impossible to know how Alexander himself would have felt about the custom.