Are there any other countries (now - or in the last few hundred years) that were/are named after their leader(s)? The only one I can think of is Saudi Arabia.
Numerous cities have been named after national leaders (often controversial and quickly changed after the death/elimination of the leader) - are there still some around?:
The obvious one - Washington.
Numerous ones once named after Stalin and Lenin.
Titograd (once again Podgorica in Montenegro (once part of Yugoslavia).
Ciudad Trujillo (once again Santo Domingo in the Dominican Republic).
Leopoldville (now Kinshasa, Congo - for King Leopold of Belgium).
Were any cities named after Hitler or Mussolini? It seems that every German (and probably Austrian) city had their “Adolf Hitler Platz”, but you never heard of a e.g. “Bad Hitler” or “Hitlerstadt”.
Rhodesia is an example of not one but two countries (North and South Rhodesia, now Zambia and Zimbabwe) named after the person who colonised them for the British Empire.
Staying in the same region, South Africa’s Pretoria was named after Andries Pretorius, founder of the Transvaal Republic, one of the predecessor states of South Africa.
Oh, and the capital of Kazakhstan, which until 2019 bore the uncreative (because it simply means “capital” in the national language) name of Astana, was renamed Nursultan, after long-serving President Nursultan Nazarbayev.
And another one (sorry for coming up with this piecemeal): Malta’s Valletta was named after a Grand Master of the Order of Malta, and thus effective ruler of the country, named Jean de la Valette. That was in the 16th century, though, so possibly outside your “last few hundred years” timeframe.
There were no cities named directly after Hitler, but the Third Reich designated a series of cities as Führer cities. It was an epithet, not a renaming, and it earmarked the city for major public construction and urban planning projects.
Over here it’s more the other way around. Windsor is the top example but dukes, for example, adopt the name of their dukedom, like Cambridge or Northumberland.
There are several towns called Lincoln is USA. Some are named after the British city of that name. Some have uncertain origin. And some, including the state capital of Nebraska, are named after the 16th president.
The Orange Free State was named after the Dutch royal House of Orange, and the Marshall Islands after the explorer John Marshall, but not sure either of those counts by the OP’s criteria. Likewise the Papal States, given that they’re not named after any specific individual leader but rather the office of their leader.
Technically the country name “Israel” is from the personal name “Israel” bestowed on the biblical patriarch Jacob, whose people became known as “Israelites”, thus originating an ethnonym and ultimately the name of a nation-state. Which I think fits the OP’s requirements if you don’t mind going back a few thousand years.
“Philippines” is derived from the name of Philip II of Spain, after whom some Spaniard (Bernardo de la Torre? Villalobos? different Wikipedia articles disagree) named the islands.
Kiribati is the modern adaptation of the name “Gilbert” for the former Gilbert Islands. But as in the case of the Marshall Islands, the name references their European “discoverer”, not a ruler or leader.
One-quarter of the well-known trivia question of naming the four state capitals named after Presidents (probably after their deaths, which means there was probably little controversy, and no real reason to name them back to whatever they were). Lincoln (Nebraska), Jackson (Mississippi), Madison (Wisconsin), and Jefferson City (Missouri).
Let’s not forget Monrovia (capital of Liberia) - named after (US president) James Monroe.
Maybe New York City would also be eligible for the OP’s criteria. It was, following the British conquest of what had previously been New Amsterdam, named specifically after James, Duke of York (subsequently King James II of England). Of course the title of Duke of York was named after the English city of that name, and to the distinguish the two the American city got the “New” added. So it is a bit fuzzy to disentangle whether the Big Apple got named after the person or the city in England.