Hm, that depends on your definition of ‘dyarchy’. (<- tl;dr)

The examples cited on Wikipedia are all over the place that there is apparently not one clear definition (maybe political scientists have one…), but the two general main approaches appear to be:

1. When the central ruling institution within an administrative structure is occupied by two people, such as the Roman consuls, or when there are two monarchs who “share” power, or sovereignty, over the state. Since shogun and emperor are qualitatively not the same, this one is out.
2. When there’s two different institutions, each headed or represented by a different person, and who have a separate, complementary, function in government, such as the government in British India established in 1919. This might be workable. (1)

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However, the early medievalists (i.e., scholars of the Kamakura period) would vehemently argue against the term: Japanese discourse on the Kamakura period state is centered around Kuroda Toshio’s kenmon taisei ron [‘Theory of a system of kenmon,’ which has very awkwardly been translated as “Gates of Power” once] and Satō Shin’ichi’s tōgoku kokka ron [‘Theory of a state in the East’]. (2)

Kuroda’s theory suggested that the medieval state as a system is constituted by all actors, which he divided into three groups, each with a distinct main function: the imperial court (“civil” administration), the shogunate (military), and religious institutions (religion, duh). All of them wielded power, all of these had estates or similar which they had more or less exclusive jurisdiction over, and we can argue a lot about how unified they were each by itself. For example, with the practice of insei government—rule by the abdicated emperor—there already was a permanent power struggle at court between the abdicated and the reigning emperor, and of course the aristocrats who manned the administration, for example. Kuroda had tried to find a model which could explain the entire medieval period, where the momentary arrangement at any given point in time was an expression of a change in the balance of power between the various constituent elements. However, he did consider the emperor the symbolic head of state.

Satō went a step further and argued that, in the Kamakura period, the shogunate was a different state, with a different” territory” (to be understood as domination over people, not land itself) over which it had exclusive jurisdiction. This means there were actually two distinct states in Japan at the time, and therefore their interaction should be classified as diplomacy. The existence of two “capitals” would support this interpretation.

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These ideas, which are the dominant accepted interpretations, would argue against the notion of a dyarchy, since in case 1, there’s only one emperor, who is considered the head of state, with a lot more than just two constituent actors present—even in aggregate, its three—, and in case 2, there’s two separate “states” to begin with, so, logically, there’s no joint government.

In the Nanbokuchō period from 1336 to 1392 we even had two imperial courts, one supported by the Ashikaga, the other opposed to them (suffice it to say, Satō’s approach stops being of relevance now). (3) For the Northern Court–Ashikaga arrangement, and the united Court–Ashikaga arrangement in Kyoto you might be tempted to use the phrase. I wouldn’t be entirely averse to it if I get a definition and explanation of the how’s and why’s (and I’m not knowledgeable enough in post-14th century politics to give a definite answer on the spot). During the Sengoku period, the territory they de facto controlled was quite small though (and did they really govern or whoever warlord just happened to have control of the capital?).

But if the requirement is that both have either the same or actually distinct governmental functions, then it becomes tricky again in the Edo period, at the latest: the emperor was, apart from nominally appointing the shogun, factually irrelevant for day-to-day governance. By this time, the shogun even represented Japan vis á vis other countries, the shogun was the top of the administrative structure which governed the country, so what function did the emperor (and his court) actually serve during this time?

From the cases given as examples of dyarchies, I am under the impression that it is strongly implied that the act of sharing power, either on de facto parity or by virtue of functional separation—and not just delegating all power to someone else—is a fundamental requirement for a dyarchy. Therefore, I at least would be inclined to say "no, not really."

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Notes:

(1) Britannica.com/topic/dyarchy

(2) Kuroda Toshio. “Chūsei no kokka to Tennō.” In Kuroda Toshio chosakushū, vol. 1: Kenmon taisei ron, Kyoto: Hozokan 1994, pg. 3–46. Essay originally published in 1963.

Satō Shin’ichi. Nihon no chūsei kokka. Tokyo: Iwanami Shoten 1983.

Amusingly, the modus operandi in most publications in the past decades is to cite both works, and then just move on, implying the “truth is somewhere out there.”

(3) Incidentally, nowadays, the Southern Court, which was the much weaker “rebel” court at the time, is considered the legitimate one by the Japanese government (back in the day it was pretty much the other way around).