It's complicated.

There's a bit of difference between (for lack of a better term) legal authority and personal authority. When society is stable and the law is strong, the two are often the same thing. But when society is in turmoil, while the two are still related, the gap between the two are often large.

In the 15th and most of the 16th century, Japan was ruled by the Muromachi Bakufu, based in Kyōto. In terms of rank at the imperial court, the Emperor was the top. The Kanpaku, often translated as imperial regent, but the documentary translated as prime minister, as the (usual) head of the aristocratic government, or court, is second. By this time, however, the aristocratic government are really only in charge of formal and/or religious rituals and officially issuing court ranks and titles. The Bakufu, the warrior government oversees Japan outside of Kyōto by appointing shugo to the provinces. These were the formal ruler of the provinces, collecting taxes, overseeing lawsuits, and mobilizing warriors for police duty and, if needed, war. Instead of the government collecting taxes from all over the country to fund its operations and pay civil servants, the Bakufu's incomes were only from the lands directly controlled by the Ashikaga family, so other warriors were under obligation to answer the calls of the government in exchange for the right to collect taxes and be in charge of plots of the provinces. Therefore, the shugo for the most part were required to stay in Kyōto or Kamakura, both to help run the government and to ensure they don't rebel. This means the person to actually oversee the province was his deputy, the shugodai, who actually assigned the warriors to carry out the tasks.

The court ranks and titles for warrior lords (leaders in the Bakufu as well as shugo of the province), including the title of Seii Taishōgun, the head of the Bakufu, are formally assigned by the court. The ranks and titles, seen as formal assignment by the Emperor and Court give much wanted and needed prestige and formal authority to the warriors. During the heyday of the Muromachi Bakufu (first half of the 15th century), however, the court basically assigns anything the Bakufu recommends. Likewise, the court basically decide among themselves who from a handful of branches of the Fujiwara families get to be Kanpaku, and the Emperor approves it. While this might seem like the Emperor, the Kanpaku, and the court were just figureheads, and in many ways they were, it is important to keep in mind that their words carried a lot of formal authority. It is also important to remember that the Bakufu also for the most part left the shugo to run the provinces, and, as mentioned, the shugo left that to the shugodai. Succession to the titles and post of the Bakufu, shugo and shugodai were also for the most part hereditary.

So by legal authority, the Emperor was the head of state and has the final say in everything. Under him, the Kanpaku leads the Court. The Emperor and court appoints the shōgun, who leads the Bakufu. The Bakufu appoints shugo, who formally rules the provinces, help run the Bakufu, and assign shugodai to actually run the affairs of the provinces. However, even at this stage during height of the Muromachi Bakufu, personal authority was incredibly important. While legally a shugo might be able to assign anyone they like to be shugodai, in practice, as the shugodai ran the province and the position was de-facto hereditary, the shugodai were able to build up personal ties through marriages or alliances, appointment and assignments of people and resources of the province, and in general just actually being there, meaning the shugodai had significant personal authority that the shugo ignores at his own peril. This was the same with the shugo and the shōgun, and even more so for the shōgun and the Court.

You might be interested in reading about the power of the shōgun and emperor through the ages that I wrote before (slightly outdated) and the hierarchies of the Japanese goverment by /u/Morricane

continued

I've not actually watched Age of Samurai: Battle for Japan, so I am going to respond to this as I know nothing how it portrays the events of the Sengoku Jidai. However, this is indeed a broad set of topics that are linked together.

Kyoto during this time is in the middle of the Muromachi Period, the later half of the Ashikaga shogunate. The Muromachi Period started when the Ashikaga ended Go Daigo's attempt to restore power to the emperor, which the Ashikaga and many samurai did not agree with. If we want to talk about the decline of Imperial centralized authority we can start here, or go back to the Kamakura shogunate, but for Sengoku Jidai we mostly have to talk about the rise of the Muromachi and the limited power the Ashikaga shoguns actually had.
(If you really want to learn about when Imperial power waned, look up information of the Kenmu Restoration, and the Northern/Southern Courts Period aka the start of the Muromachi and are interesting stuff)

You see imperial power and the emperor had local lords, usually in the form of imperial aristocrats rule imperial territories. These imperial estates granted local power to bureaucrats and distinguished families. There were also military postings in said territories where samurai families would slowly grow local power. However, by the time of the Onin War near the middle of the Ashikaga's reign. The local lands that had been administrated by bureaucrats who were given authority from the capital slowly lost those lands. These local imperial bureaucrats were replaced by military governors. The Ashikaga gave land to samurai and powerful military families and didn't control a lot of land themselves. Those same local samurai families. Whom, would later become daimyo families that we will see in the Sengoku Jidai.

As time went on the local daimyo had more power and the the Ashikaga shoguns had to rely on them to maintain power. The Ashikaga meanwhile used the emperor for their own legitimacy. The emperor by the time of the Onin War was destitute and Kyoto and the emperor were only symbolic. Local power and conflict began to start small scale wars, hence why the Sengoku Jidai is known as the "Warring States Period". As many of these local areas had more authority than the capital, the emperor or the Ashikaga.

Many of the powerful family before the Onin War had also been granted imperial country under the shoen or estates. The Uesugi, Imagawa, etc held prestige because they claimed not only military power, but Imperial power. Although, locally the only power that meant anything was military. We slowly see the erosion of Imperial and Shogun rule by the time Nobugana comes to power, and it is with this that military power reigns absolute and any use of Imperial power is just for prestige, power, and legitimacy.