You are probably right, John R, it defies logic or, at least, the rules, except for those who might argue that as the fount of honours HM can do what he likes (I am not amongst them).

My interpretation, and that is all the following is, is that, following the then Aga Khan's removal from Persia to (British) India, and his support for British forces in Afghanistan, the British Government was happy to accord him Princely status in India, even though he was not a territorial ruler. As Princes of the the Qajar Imperial House of Persia (through their descent from a daughter of Shah Fath Ali Khan), they were already titled 'Prince/ss". and as influential Shia religious leaders, the Aga Khans (Agas Khan?) were valuable allies for the British in Muslim India, and beyond. They were accorded the honorific of HH so that their status among the many, often less important (to the British) Indian Princes was recognised.

Aga Khan III's relocation to Europe and the absence of a territorial presence in Princely India left the family with Princely status not tied to independent India but with an historical relationship with the British Crown,

The late Queen's granting the last Aga Khan the HH honorific, and The King similarly according that style to Prince Rahim, might be seen as survivals from an earlier, Imperial age or, as The King has argued, a recognition of the long and important relationship between the two families. It also might reflect the fact that, although the Aga Khan and his family are British citizens, he is treated very much as the equivalent of a Head of State.

But, as I agreed at the start, this is all quite illogical according to the formal rules. But then, as Lucas noted below, so was Princess Marie Louise (and, for that Princess Victoria of Hesse and by the Rhine, Countess of Milford Haven).