Greetings! This in indeed an interesting question, as with a title as grand as Deutscher Kaiser (German emperor) and (at least in the time of Wilhelm II) "All Highest Warlord", it seems logical to assume that the Kaiser was politically without equals in the German Empire. However, upon closer inspection of the constitution and some secondary literature from the era, we can actually discover that he was less powerful than the President of the United States, as OP has suggested in their explanation of the question. Let's begin.

Under Article 11 of Section IV of the Constitution of the German Reich (1871-1919), otherwise known as the Constitution of the German Empire to avoid confusion with the one pertaining to the successor state; the Weimar Republic, one finds the following:

Article 11

The King of Prussia shall be the President of the Confederation, and shall have the title of German Emperor. The Emperor shall represent the Empire among nations, declare war, and conclude peace in the name of the same, enter into alliances and other conventions with foreign countries, accredit ambassadors, and receive them. For declaration of war in the name of the Empire the consent of the Federal Council shall be required. Except in case of an attack upon the territory of the Confederation or its coasts.

So far as treaties with foreign countries refer to matters which, according to Article 4, are to regulated by the Legislature of the Empire, the consent of the Federal Council shall be required for their ratification, and the approval of the Diet shall be necessary to render them valid.

Here we have an interesting glimpse into the real authority invested into the Kaiser by the constitution. Under this constitution, the Kaiser acted as president of the Confederation (the twenty five member states which composed the German Empire). In his capacity as Bundespräsidium (Federal Presidency), the office of the King of Prussia was directly tied to the office of the President of the Confederation, and was accorded the title Deutscher Kaiser from this duty. Within the constitution, the Kaiser was technically head of the Bundesrat, or Federal Council. This council of 58 representatives from each of the 25 member states (later 61 representatives from 26 member states, with the addition of Alsace-Lorraine as a represented polity from 1911 onwards) was the highest authority of the German Empire.

Now, alongside the Bundesrat, the German Reich was composed of a Reichstag (or Imperial Diet), which consisted of 382 (later 397) elected representatives from the various states (and later political parties) of the German Empire. Here the Kaiser was also, in technical terms at least, one among equals. He (unlike the President of the United States), did not have the power to veto laws which he did not agree with, and under the constitution any imperial laws which he wished to enacted would have to be co-signed by the Chancellor of the Empire (or Reichskanzler) to be legally binding and enforceable. However, in reality, Kaiser Wilhelm II exercised a fair degree of influence (but not control) over the Reichstag, in part due to his own public prestige amongst the German peoples, but also in part due to the fact that the Reichstag, owing to its heterogeneous composition of rival political parties, could never pose an immense threat to the Kaiser's authority. The following articles below give proof of this hidden influence:

Article 17

To the Emperor shall belong the right to prepare and publish the laws of the Empire. The decrees and ordinances of the Emperor shall be published in the name of the Empire, and require for their validity the signature of the chancellor of the Empire, who thereby becomes responsible for their execution.

Article 18

The Emperor shall appoint the Imperial officials, require them to take the oath of allegiance, and dismiss them when necessary. Officials appointed to any office of the Empire from one of the States of the Confederation shall enjoy the same rights as those to which they are entitled in their native States by their official position, provided no other legislative provision shall have been made previously to their entrance into the service of the Empire.

And then, we get onto the actual control the Kaiser possessed, and if one scrolls far down enough on the constitution, they find this sliver of legislative gold:

Article 63

The total land force of the Empire shall form one army, which, in war and in peace, shall be under the command of the Emperor.

The same applied to the Navy, and thus the Kaiser (with the help of his "Military cabinet" and "Navy cabinet") practically controlled the armed forces of the German Empire. This authority implies something interesting: although the Kaiser would require (in law) the consent of the Bundesrat and the Reichstag to enact an offensive war, how and where that war was fought was entirely under his technical control. Neither parliamentary assembly could veto his deployment of the armed forces, and neither could exercise any influence over the various offensives and campaigns which the Kaiser commanded. Further still, Wilhelm II was notorious for shaping the position of Reichskanzler into a figurehead, using them merely as a mouthpiece to direct the foreign and domestic policy of the Empire to his will. With the dismissal of Otto von Bismarck in 1890, his successors had to deal with the ambitions of Wilhelm II, and many resigned or were dismissed if they could not follow along with the Kaiser.

These final two quotes from Wolf von Schierbrand (writing in 1902) and Philip Zorn (writing in 1899) illustrates how the reality of the Kaiser's authority actually subverted (and at times were directly in violation of) the constitution which he was bound to uphold:

"These autocratic leanings of his [Wilhelm II] would, in other countries with a longer past of constitutionalism, be a more dangerous defect. But in Germany, where parliamentarism is an importation, which has by no means as yet been as firmly established as in England, France, or even Italy, these absolutistic tendencies of the Emperor figure not so prominently in the people's eyes as one might think. For what in Germany is termed the "monarchic principle" is rooted very deeply in the heart of the nation as a whole.

"The German Emperor has the rights of a monarch toward the parliamentary representation of the German people, the Reichstag.

The German Emperor could under certain circumstances constitutionally suspend the operations of the Bundesrath [note: the additional "h" at the end was added as part of the 1902 spelling reform].

He has monarchical power in the appointment of the executive organs of the Empire the chancellor, the secretaries of state and the other imperial officials.

On the contrary, the Emperor has no monarchical power in the legislation of the Empire, for he lacks the right of approval and an independent right of issuing decrees. In the Bundesrath Prussia, with its seventeen votes, faces forty-one votes of other states; it can always be outvoted except in matters concerning army, navy, customs duties and indirect taxes."

Hopefully this brief overview helps, and feel free to ask any further questions on the role of the Deutscher Kaiser within the German Empire.