As far as I'm aware, the vast majority of countries in asia were Monarchies of some form or another. During the Victorian Era there was generally an increase in contact between the east and the west; the British Empire facilitating travel and trade between multiple countries and so on.
Also during this time you had things such as Admiral Perry forcing Japan to open to trade, and America conquering the Philippines. What I want to know is, is there any record of Asian countries reacting to the U.S.A.'s government? Do we have any notable incidents where, say, Japan or Malaysia attempted to inquire about America's "King"? As a bonus question: was the concept of democracy overall pretty foreign to Asia around the Victorian Era? Was there any consensus on whether it was a good or bad form of government by the people of that region?
In 1849, a Chinese official and geographer by the name of Xu Jiyu wrote a world geography book called A Short Account of the Maritime Circuit where he described each countries around the world. He had opportunities to interact with many foreign officials and missionaries where he learned a great deal about other nations. Xu was known for his respect for western education and political systems. He described British Parliamentary system as (my translation):
"...a meeting house, divided in two. One room is full of aristocrats and teachers of the Christian religion. The other is full of men of knowledge endorsed by the people. If the kingdom needs to make a decision, the aristocrats will be informed and make a decision, it is then passed to the house of commoners for their approval otherwise the matter will not be discussed further. If a commoner would like to solve the ills of society, he shall take the matter with the house of commoners, then he will be passed to the aristocrats, if he meets their approval then the king will be informed. If a commoner would like to make an accusation, the same process are to take place...this system is used in countries all over Europe, not unique to England..."
His description of George Washington:
"Washington is an impressive person. He fought for independence with his people, more brave than Chen Sheng and Wu Guang, more effective than Cao Cao and Liu Bei. After independence, he has acquired control over a large territory but he still refused to establish a dynasty or pass his title to his children for the sake of morality and justice. He made the precedence of refusing more than two terms."
(The paragraph on Washington is carved in the Washington Monument.)
FYI Japan's main European trading partner during the Tokugawa era with the Dutch Republic, which was a, well, republic. This is where they got the majority of their information about the west from during that era, and as such it is highly unlikely they would have been unaware of republican forms of government.