Hey!

This is an interesting question and a common misconception. Simply put, the Qurʾan itself does not forbid the representation of Muhammad or other figures. There are some hadith that are cited against it, such as Sahih al-Bukhari 7.834 and 7.838, which both claim that the "picture makers" or those who "try to make the like of Allah's creation" will receive the severest punishments on the day of judgement. This is often interpreted as a general ban on depicting Muhammad (and other humans, animals, etc., hence why much of Islamic art is based upon plants and geometric patterns).

However, this does not mean that depiction of Muhammad never existed in the Islamic world. In fact, at certain times they abounded. For example Christiane Gruber writes that "illustrations of the Prophet flourished in Persian lands, especially during the Ilkhanid (1256-1353), Timurid (1370-1506), and Safavid (1501-1722) periods". Hafiz-i Abru, the painter of the portrait you saw, was a Persian working in the Timurid court in Herat, and so it is not historically inaccurate for him to have depicted Muhammad.

So how can Islamic artists justify depicting Muhammad?

In the article "Blasphemy and Violence", Ron E. Hassner writes that

The Qur'an prohibits idolatry but has little to say about figural representation and formal jurisprudence is indifferent but not hostile to human images in art (Grabar 1977:45-47). Rather than prohibit images, Islam, like most religious movements, limits the types of images that can be used and the manner in which they can be used. It is the underlying meaning of a representation, not the fact of representation as such, that is potentially offensive to Muslims.

It is the fear of conflating iconography and idols that is the reasoning behind the ban. The sin of shirk, or assigning partners to Allah (and thus denying Allah's unity) is the most unforgivable sin in Islam. This is why you will not see representations of Muhammad inside Mosques, where they may become the object of worship instead of Allah himself.

However, this injunction does not apply to private collections, which were relatively common from the 12th to the 14th century before declining in popularity. The painting you saw, from 1430AD, was right at the tail end of this period. When they existed within private collections, particularly within courts, these depictions were not at risk of becoming the object of idolatry.

When you do find depictions of Muhammad, more commonly in Shiʿa or Ṣūfī works, they tend to come as historical illustrations and are not viewed as idols. That appears to be the case with this painting, as it is from a manuscript depicting the history of Muhammad receiving the Revelation.

Sources

Hassner, Ron E. “Blasphemy and Violence.” International Studies Quarterly, vol. 55, no. 1, 2011, pp. 23–45., www.jstor.org/stable/23019512.

GRUBER, CHRISTIANE. “BETWEEN LOGOS ( KALIMA ) AND LIGHT ( NŪR ): REPRESENTATIONS OF THE PROPHET MUHAMMAD IN ISLAMIC PAINTING.” Muqarnas, vol. 26, 2009, pp. 229–262., www.jstor.org/stable/27811142.

If you wanted to see some more depictions of Muhammad in Islamic art, "From the Literal to the Spiritual: The Development of the Prophet Muhammad's Portrayal from 13th Century Ilkhanid Miniatures to 17th Century Ottoman Art" by Wijdan Ali talks about how perceptions of depicting the Prophet changed and provides many example paintings at the end.