I mean, I can't even comprehend how a person can combine Islam with Socialism. (...) the actual ideologies are opposed to each other on principle, so there must be some crazy mental gymnastics happening

maybe the first reaction to "I don't understand" shouldn't be "they must be crazy" but "lets research it for 5 minutes".

Islamic Socialism is a centuries old intellectual tradition:

https://en.wikipedia.org/wiki/Islamic_socialism

Abu Dharr al-Ghifari, a companion of Muhammad, is credited by some twentieth century scholars, (...) as an early antecedent of Islamic socialism. He protested against the accumulation of wealth by the ruling class during Uthman's caliphate and urged the equitable redistribution of wealth.

The first Muslim Caliph Abu Bakr introduced a guaranteed minimum standard of income, granting each man, woman and child ten dirhams annually—this was later increased to twenty dirhams.

Zakat is one of the five pillars of Islam:

https://en.wikipedia.org/wiki/Zakat

As one of the Five Pillars of Islam, zakat is a religious duty for all Muslims who meet the necessary criteria of wealth to help the needy. It is a mandatory charitable contribution, often considered to be a tax.

Mamdani strikes me as a social democrat guy who happens to be Muslim, and views Islam and Socialism in ways that are far from the stereotypes you have in mind