The short answer is that your question rests on a flawed assumption: the Qurʾan does not prohibit figural representation, be it of human beings in general or even the Prophet Muhammad specifically.

What the Qurʾan does forbid is idolatry and the production of idols, which certain hadith have interpreted as a ban on human representation more broadly. Even these hadith, however, are themselves open to interpretation. Take, for example, the following relation, which seems to be as strong a condemnation as you could ever find of portraiture in Islam:

Ibn ʿAbbas (May Allah be pleased with them) said: I heard the Messenger of Allah (ﷺ) saying, "Every painter will go to Hell, and for every portrait he has made, there will be appointed one who will chastise him in the Hell." Ibn ʿAbbas said: If you have to do it, draw pictures of trees and other inanimate things.

And yet it is this same hadith that Muhammad ʿAbduh, Grand Mufti of Egypt from 1899 to 1905, cites in a famous fatwa permitting figural art! ʿAbduh reasons that this hadith does not apply, since it refers to painters and portraiture in the days of paganism, when such works might have been worshipped as idols:

Images were taken up during this time for two reasons: the first was distraction and the second was to seek blessing from the image of whichever of the righteous ones is depicted; the first [reason] religion detests and the second Islam came to wipe out. The painter is in both cases a distraction from God and a facilitator of polytheism. If these two obstacles are removed and the benefit pursued then the depiction of people is the same as the depiction of plants and trees or objects.

Since there is no danger of good Muslims lapsing into the idolatrous worship of figural art, there is no harm in permitting such art in an Islamic context. While ʿAbduh's fatwa was (and remains) far from uncontroversial and was issued as part of a larger project of spiritual reform, I mention it here as an example of how jurists could legitimate the depiction of the human form using the same texts that were elsewhere used to condemn it.

The history of figural art in Islam predates Muhammad ʿAbduh, of course; it also predates the Ottomans. The remains of the late Umayyad (mid-eighth century) palace of Khirbat al-Majfar show some relatively early examples of Islamic sculpture and mural work, for example—including a figure widely held to be a statue of the caliph himself!

The tradition that most influenced Ottoman miniaturism and portraiture, however, is that of manuscript illustration. Medieval Persian manuscripts—whether literary (the Great Mongol Shahnama) or historical (Rashid al-Din's Jamiʿ al-tavarikh) or even religious (the Ilkhanid Miʿrajnama)—used figural art to great effect and developed a rich tradition of portraiture. The examples in parentheses are all from the Mongol period, because few illustrated Saljuq examples survive to this day—but Saljuq manuscript arts influenced other pictorial traditions, such as the ceramic genre of minaʿi ware, where you can see similar illustrative techniques. The Ottomans therefore had no reason to think there was anything particularly un-Islamic about royal portraiture, a practice with noble precedents in Iranian art.

Perhaps this changed by the eighteenth century or so: Ignatius Mouradgea d'Ohsson reported that the Ottoman sultans of his time [the 1780s] kept the imperial portrait collection hidden from not only the public, but also courtiers who did not enjoy the sultans' confidence. But from there, of course, it's a mere century and a quarter to Muhammad ʿAbduh! All in all, painting has probably been viewed as licit more often than not in the Islamic world, especially in those areas that looked to Persian rather than Arab culture for their guide.

READING

Christiane Gruber has written quite a bit about the history of figural representation in the Islamic world. See, for example, her "Idols and Figural Images in Islam: A Brief Dive into a Perennial Debate" (or, for a less academic piece, "The Koran Does Not Forbid Images of the Prophet.")

On Muhammad ʿAbduh's fatwa and its cultural/religious context, see Dina Ramadan, "'One of the Best Tools for Learning': Rethinking the Role of 'Abduh's Fatwa in Egyptian Art History"; and Stephen Vernoit, "The Visual Arts in Nineteenth-Century Muslim Thought," in Islamic Art in the 19th Century: Tradition, Innovation, and Eclecticism, ed. Doris Behrens-Abouseif and Stephen Vernoit (Leiden: Brill, 2006), 19-35.