I believe Singapore is not the monarchy people perceived.
Why people say Singapore is a Monarchy(Lee Dynasty) is because Lee Hsien Loong is Lee Kuan Yew's son (male line, Y-chromosome). When Lee Kuan Yew retired (seen as abdicated) Lee Hsien Loong ascended with Goh Chok Tong as seat warmer in between.
What make Singapore not a monarchy is PM do not make law as he wish like monarchy do. In Singapore, law need to go through legislation for debate to be gazetted. Although PAP won majority all the time and PM have big influence on the law making process but he is not absolute. Not even like US President where he can issue Executive Order, Singapore PM can't.
Singapore PM do not have the legal immunity or at least not at the same level as monarchy do. One of the distinct characteristics of monarchy is legal immunity.
Singapore PM is chosen by merit. Not only he need to be competent but also corruption-free and having high moral virtues. This is not require for monarchy. Whereas, traditionally monarchy is purely pass down by bloodline. Their successor need not be chosen because of education achievement and moral virtues. They are either pre-determine by monarchy tradition or favour from this parents.
Also because of the above, monarch normally delegate their administration role to professional who competent to handle nation affairs even they hold title like Prime Minister, Chef Command of Army and President. Whereas, Singapore PM and Ministers were chosen for their competency and expertise that they are there to perform.
https://youtu.be/cn_nJmMoO6gOf
course, there are monarch system that have election by elites to chose the King. Malaysia Negeri Sembilan is one of the kingdom where Sultan(King) is chosen by elites where competency, personal quality and a list of criteria is assess from a few of the royal bloodline.
Sweden monarch is voted by people through referendum.
Source:Elective monarchy - Wikipedia (Elective monarchy - Wikipedia)
North Korea is calling itself republic democracy(DPRK, Democratic People's Republic of Korea). Kim was elected and election was held where he was voted by the people but the country is no where near democracy. The throne belong to Kim family since independent. From his grandpa to papa to him. There's no doubt it is a Kim dynasty. It is more Communist than any country on earth now.
Whether Singapore is Monarchy or not is not important, more important is the people is having good life. Only wealthy nation wealthy people have freedom of choice. I don't think Singapore has a monarchy. Lee dynasty is just what local folks talks. Even it is a monarchy it is not a problem as long as the country is well managed.
So it is all labeling. There's no absolute democracy and Communist but surely there's some absolute monarchy country.