As I was thinking about the different commonwealth realms sharing a monarch, I realized that except for the UK, no Commonwealth Realm has decided to name its state "Kingdom of X", this, even though they are quite clearly kingdoms. Looking more into it, I also realized that the majority of them have no long formal name at all, although those that do tend to be Commonwealths/Federations. The vast majority of sovereign states have long form names, so the fact they dont follow this global trend made me think this is also tied and possibly part of the reason none have named themselves a kingdom. So why is this the case?
As you note, most countries have long form names. These usually described the type of government - for example, Kingdom of Sweden, Kingdom of Spain, Kingdom of the Netherlands, Republic of Ghana, Republic of Kenya, Republic of Latvia. Some add other adjectives, such as "Democratic", "Islamic", etc. There are some "government-neutral" names, which don't specify the specific form of government: United Mexican States, Independent State of Samoa, Swiss Confederation.
This "neutral group" includes the majority of the long-name Commonwealth realms: Commonwealth of Australia, Federation of Saint Christopher and Nevis, Commonwealth of The Bahamas, Independent State of Papua New Guinea. There are two long-name exceptions to this: the UK, and Dominion of Canada. While "Canada" is the legal name of the state, "Dominion of Canada" is still the official long form of the name.
The UK, as the core of the former Empire, is a special case. Canada is the clue to why the other long-name realms have "neutral" names. Since the realms are not republics, names like "Republic of Canada", "republic of Australia", etc. were out. Why not "Kingdom"? There were multiple proposals of "Kingdom of Canada" (and, more rarely, "Republic of Canada") as the Canadian provinces were being united in the 1860s (with Newfoundland to join much, much later). The British Colonial Office rejected "Kingdom" - one consideration was not wanting to offend the southern neighbour who had invaded Canada twice before (i.e., the USA). As a result, the more neutral "Dominion of Canada" was chosen. The choice for the later dominions/realms was simple: no long name, a government-neutral long name, or "Dominion of <foo>". New Zealand became, a few decades later, "Dominion of New Zealand", but dropped the "Dominion" later, and the other dominions/realms (which mostly gained independence in the 2nd half of the 20th century) went short-name-only or "neutral".
The idea of the realms being called kingdoms persisted, as shown by this 1910 article (almost a half-century after the British North America Act (1867) which established the Dominion of Canada):