Honestly, I have no idea.  It’s certainly not any type of theological interest or connection.  Unlike the Presidents, the Popes should be as foreign or distant to me as the examples you used — the Japanese emperors or Spanish monarchs — and, really, they are.  Yet, those who have reigned as Pope and the Papacy itself definitely piques my interest in a way second only to the American Presidents and American Presidency.

The best explanation that I can give you for my interest in the Popes and Papacy despite the fact that I am a committed non-believer actually comes from a Catholic priest.  In The Making of the Popes 1978, Father Andrew Greeley differentiates between “papal mystique” and “papal policies”.  It’s that mystique that draws me in — an interest in the people who put on the robes and the funny hats and become this incredible powerful, absolute monarch who is not only a supposedly infallible religious leader to over a billion human beings worldwide but also the temporal leader of a sovereign state with borderline dictatorial powers.  And, most amazingly of all, there is no Biblical or secular basis for this hugely powerful position.  

As Father Greeley writes about the “papal mystique”: “You can come and shout ‘Viva il Papa’ at St. Peter’s because of the symbolic power of the office and still not really take seriously the kind of stuff the Pope tries to teach you.”  When I’m studying the Popes and I read their encyclicals or their motu proprios, it’s not a religious thing to me — it’s filling in the biographical blanks.  The Pope’s power is supposedly derived from God.  I don’t believe in God.  But the Pope indeed has power.  So the mystique for me becomes a search for where that power actually comes from, who truly wields it, what kind of person does it take to become and reign as Pope, and so on.  That’s where my interests in the subject lie, I guess.