I'm slightly updating an answer I wrote a while back on why some medieval Muslims felt that only certain types of alcohol were banned (e.g., wine and beer, but not mead).

There been a long-running debate in schools of Islamic jurisprudence over what qualifies as khumur ("intoxicants," singular khamr), which are explicitly forbidden by both God in the Qur'an and the Prophet Mohammed in the Hadith.

On one side were the Maliki and Shafi schools, with their roots in Arabia, which advocate the belief that all alcoholic beverages are khumur and are therefore forbidden. This goes back to the days of the Maliki school's founder, Malik ibn Anas (711-795), who cited scriptural evidence for general prohibition based on the Qur'an and Hadith. Later Maliki scholars like Sanhun al-Tanukhi (c. 782-855) would say that the prohibition applies to all drinks with even the capacity to ferment, though that idea had little traction in future generations. Similarly, the founding Shafi scholar Idris al-Shafi (767-820) argued for general prohibition based on the traditions of early Islamic communities. Later Shafi scholars like al-Mawardi (972-1058), al-Baghawi (1041-1122), and al-Rafi'i (1160-1226) focus largely on using their claims to counter the Hanafi stance.

On the other side was the Hanafi school, with its roots in Iraq, which argued that only grape and date wines (the two alcoholic beverages explicitly mentioned in the Qur'an and the Hadith) are forbidden. They also claim that that it is not drinking per se that is the sin, but drunkenness, citing verses such as one from the Hadith where Muhammad condemned the "last cup which intoxicates." This idea of "narrow prohibition" started with the school's founder, Abu Hanifa (699-767), based on his literal interpretation of Quranic verses on khumur. Many of the debates within the Hanafi asked whether it only applies to only grape & date wines, all grape & date juice drinks, or only certain methods of production or fermentation. Starting in the 12th century with scholars like al-Marghinani (1135-1197), Hanafi scholars began declaring broader and broader categories of alcohol unlawful. By the 14th century, the belief in general prohibition of alcohol was commonly accepted, and works of earlier Hanafi founders were reinterpreted to keep things ideologically consistent.

Why did the former stance eventually win out over the latter? As the 12th century Maliki scholar Ibn Rushd (better known in the west as Averroes) points out in his treatise Bidayat al-Mujtahid (Primer of the Discretionary Scholar), the Hanafi school had a weaker justification. There's essentially a hierarchy of precedence when it comes for determining Islamic law: explicit commands from God in the Qur'an > explicit commands by Muhammad in the Hadith > qiyas (analogical reasoning) to determine implicit laws in scripture > ijma (scholarly consensus). And the schools in favor of general prohibition (especially the Malikis) built a far stronger case based on scripture than the Hanafis. There were also likely social stigmas at work, with consumption of alcohol becoming increasingly associated with pre-Islamic or foreign societies, and the Hanafis increasingly seemed like the odd ones out.

Since the Hanafi school was and largely still is the dominant school in Anatolia, the Levant, Iraq, Egypt, and Central Asia (as well as that endorsed by Islamic kingdoms like the Abbasids, Seljuks, and Khwarezmians), not to mention how Rumi himself was a Hanafi jurist, it is perfectly reasonable that people living in such a place and time would believe in a far laxer prohibition on alcohol than one we commonly associate with modern Islam.

My main source was "Contesting Intoxication: Early Juristic Debates over the Lawfulness of Alcoholic Beverages" by Najam Haider in Islamic Law & Society (which can be read here), with some supplementary info from John Esposito's book Islam: the Straight Path.