The argument raises real questions, but it does not establish that the Quran borrowed a non-historical Joseph story. The distinction between Joseph's ruler being called "king" and Moses's ruler being called "Pharaoh" is suggestive, yet neither its probability nor its historical force should be exaggerated.
What the criticism gets right
The critic is right that the commonly presented "miracle" is not a mathematical proof. The Quran mentions Joseph's ruler only a limited number of times, and a writer could avoid the word "Pharaoh" by chance, especially since "king" is a perfectly natural word in a narrative. The probability estimate, however, depends on assumptions about how often each term should have been chosen. Those choices are not independent coin tosses: the Joseph and Moses narratives use different settings, characters, and narrative traditions.
The critic also points out that Fir'awn is grammatically treated in Arabic as a proper name, appearing alongside names such as Haman. That differs from the Egyptian expression meaning "Great House," which became a royal title in the New Kingdom. Other proposed difficulties include crucifixion, the word darahim for payment, the mention of rain, and Haman's association with Pharaoh rather than with the Persian court in Esther.
What the wording actually establishes
And the king said, "Bring him to me; I will appoint him exclusively to myself."
Quran 12:54
And the king said, "I do not see you as anything but a sorcerer."
Quran 40:23-24
The Quran consistently calls Joseph's ruler al-malik, while it consistently calls Moses's ruler Fir'awn. That fits the broad historical distinction that "Pharaoh" became a conventional royal designation later than the era usually associated with Joseph. It is therefore a reasonable piece of cumulative evidence, not a standalone demonstration.
Calling Fir'awn a proper name in Arabic does not prove the Quran claims that it was an Egyptian birth name. Foreign titles commonly become fixed names in another language. "The house of Pharaoh" is also intelligible as the household or regime of the ruler, just as a title can generate ordinary expressions. Classical commentators generally understood Pharaoh as the title of Egypt's ruler, even while Arabic grammar treated the word as a name.
The other alleged anachronisms
These points are less decisive than they are often made to sound. Crucifixion or public suspension was not restricted to one later civilization. Darāhim can describe silver coins or pieces in the language of the Quran's audience without claiming that a modern coin system existed in Joseph's Egypt. Quran 12:49 mentions abundant rain and relief; it does not say that Egyptian agriculture normally depended on rainfall rather than the Nile.
Haman is a genuine point for comparison with later Jewish literature, but a similar name does not prove literary borrowing, and the Quran does not identify its Haman with the figure in Esther.
The strongest conclusion is modest: the Quran's terminology is compatible with a historically informed distinction and creates a meaningful question for critics. It does not by itself prove the story's historicity, nor do the alleged anachronisms by themselves prove invention or borrowing. The case must be judged from the whole Quranic narrative, its language, its theology, and the competing historical explanations.