I understand that Hamadani was born into a Jewish family and later converted to Islam and that he was born under Mongol rule. May this have anything to do with it? Or is it more simple, like a misunderstanding of how aniconism works in Islam?
Edit: spelling
Hey! I wrote about depictions of Muhammad in a previous (and now that I have looked at it, quite old) answer here. I made slight edits to the below, and am happy to try to answer any follow-ups if I have the time.
In short,. Aniconism isn't strictly forbidden in Islam. Indeed, its opposition was least in Shi'a communities, and particularly in Shi'a Persia (where Rashid al-din Hamadani was from) [Edit to add: I was kindly reminded that the Shi'a conversion of Iran happened later—roughly 1500 AD. The pre-Shi'a culture of Persia was nonetheless still more accepting of depicting creation than the Levant]. The main prohibition is on idolatry, and manuscripts in private collections were seen at less risk of contributing to idolatry than, e.g., depictions within mosques themselves.
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) [Note: this is the period within which Hamadani lived], Timurid (1370-1506), and Safavid (1501-1722) periods".
Hamadani himself was born—as his name may indicate—in Hamadan (modern day Iran) under Mongol rule. Given the time period and the influence of Mongol rule itself (if I recall correctly, Jami'a al-Tawarikh was commissioned by the Mongol ruler), it is not historically inaccurate for him to have depicted 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 manuscripts you saw were produced in the early 1300s AD, in the center of this period. When such manuscripts existed within private collections, particularly within courts or other institutions of learning, these depictions were not at the same 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. Jami'a al-Tawarikh was indeed a historical work, despite Persia not yet being Shi'a.
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.