Short answer:

Japan has a political history with both real and superficial parallels to other polities, but in general analogies are more trouble than they're worth. Treat Japanese political terms and institutions as their own thing and they'll be easier to understand. Whatever European parallel you think you see . . . is often too simple.

Discussion:

You should think of the Japanese Emperor as a distinct institution, and understand his changing role in the context of Japanese history. Japanese Emperors have had considerable temporal power at times. Analogies typically obscure more than they enlighten; you'll find medieval and renaissance commentaries that analogize the Caliphs of Islam to Popes . . . does that tell you much? Courtly Japan can be understood in universal terms: it's a struggle for power. And that struggle for power takes place over a remarkably complex and historically specific landscape of political and religious institutions, over a backdrop of clan rivalries and personal ambitions that have to be understood as the people who lived them understood them.

Japanese Emperors had real power, they substantially lost it in the 13th and 14th centuries, but it's much too simple to say "the Emperor is a puppet". There are all kinds of complexities, including "retired Emperor". Indeed, you have on occasion political battles _between_ retired Emperors.

It's also worth noting that there were Popes who had considerable temporal power, or they were allied to those who did, and pontiffs who had no temporal power often had (and continue to have) substantial political power . . . so they're politically more powerful than, say, the Emperors of the Tokugawa Shogunate. The religious role of the Popes is also quite different: to a great extent the Emperor's religious role are the ceremonies he performs, and although there are specific religious institutions associated with the Imperial family, the Japanese Emperor didn't "run the religion" in the way that that Popes do and have done.

"The Shogun was the real leader."

Sometimes, sometimes not. Japan has a history of veiled power, of regents and Shogunal regents. That is, not only on occasion do you have an Emperor without much power, you actually have periods where the Shogun didn't have much power either, and a third figure actually ruled. There's a title that's less known, the Shikken ( 執権), the Shogun's regent. This occurs during a period known as the Shikken Seiji -- the rule of the Shogunal Regents, during the Kamakura period.

All of this is fascinating material for folks looking for a baroquely complex "Game of Thrones"; none of it is made any easier to understand by analogy to Europe and in particularly to European religious institutions that really don't parallel Japan at all.

To understand Japanese history and institutions, its best to take them as they are. The Japanese Emperor doesn't even track all that closely to the Chinese Emperor in terms of who he was in the society. As you dig into Japanese court politics, you find categories of influence that are specific, for example the role of the menoto, the wet-nurse; she and her family had a connection to her milk-child that was often of great influence. Wet nurses themselves give rise to important courtly clans.

So with Emperors, imperial regents, retired Emperors, Shoguns, Shogunal regents, and more -- the specificity of Japan's political arrangements demand that you treat it as its own thing to gain any understanding. Whenever foreigners look at Japan and draw analogies "well, the Samurai are like European feudal knights" -- well, they are . .. and then again, they're not. It's better just to say "daimyo and samurai" rather than to worry about just how the relationship does/doesn't map to the Counts of Anjou.

​

See:

Bauer, Mikael. “Conflating Monastic and Imperial Lineage: The Retired Emperors' Period Reformulated.” Monumenta Nipponica, vol. 67, no. 2, 2012, pp. 239–262.,

Spafford, David. “Emperor and Shogun, Pope and King: The Development of Japan's Warrior Aristocracy.” Bulletin of the Detroit Institute of Arts, vol. 88, no. 1/4, 2014, pp. 10–19.

Jeffrey P. Mass, Yoritomo and the Founding of the First Bakufu: The Origins of Dual Government in Japan (Stanford, CA, 1999)

Conlan, Thomas D. “Thicker than Blood: The Social and Political Significance of Wet Nurses in Japan, 950-1330.” Harvard Journal of Asiatic Studies, vol. 65, no. 1, 2005, pp. 159–205.