The following types of subject all count for this achievement:
Note, however, that trade companies do not count.
Marches, vassals, personal unions and client states all count towards a player's diplomatic relations cap, so having 21 of these in any combination would leave one unable to earn diplomatic power and therefore unable to annex any of them. Colonial nations and protectorates, on the other hand, do not count towards this limit, making them a natural choice for this achievement.
There are several possible ways to approach this achievement.
Mandate of Heaven
The achievement is easy to get playing as Ming or any other Chinese Emperor with the Mandate of Heaven DLC enabled. Colonizing single province islands around the world will never give a penalty to mandate, while still providing you with bases that can be used to obtain tributaries with. In order to establish a nation as a tributary state, a player can secure a border with the nation they plan on establishing as a tributary, possibly by establishing a colony bordering a province of the nation. Alternatively, war can be declared on said nation to force tributary in the peace deal, either with a conquest CB or declaring a no CB war if necessary. Once a nation is established as a tributary state, the player can border said tributary without incurring a mandate penalty, making future tributary opportunities easier.
Exploration ideas are recommended to allow for the colonization of provinces and exploration, although influence ideas can also be taken to reduce liberty desire on tributary states and make it easier to diplomatically establish tributaries. It is not a good idea to establish a nation as a tributary if they are likely to be attacked by a large nation(such as the Ottomans) as that will force the player to fight a war against them. Each tributary must have 5 or more provinces in order to count towards the achievement, so nations that are smaller should be ignored unless they expand to the necessary size. Due to the nature of tributary states it is possible that a tributary state that was already established with more than 5 provinces could end up with less than 5 provinces, so it is important to keep that in mind.
The colonial method
For this method, it is best to play as a strong colonial nation such as Portugal or preferable Castile. By taking exploration ideas and then expansion ideas as first groups, a player can quickly colonise the Americas and amass many colonial nations in the process. There are 11 different colonial regions in the Americas (and a 12th in Australia) and the goal is to have a colonial nation in as many of the regions as possible. As such, it is best to settle enough colonies in each region to form the colonial nation, and then move on.
While colonising, it is also advisable to be on the look-out for nations that can be turned into protectorates, both in the Americas and in West Africa. Since the achievement requires that the subject nations have at least 5 provinces, be sure to only take protectorates that are that big, or that you can make that big by giving them provinces. Also be wary of taking any protectorates that are considerably larger than that, since larger, more powerful subjects will have higher liberty desire.
It is important to use diplomats throughout this process to continually improve relations with all the subject nations, since good relations decrease liberty desire.
Depending on the specifics of a game, this combination of colonial nations and protectorates (and possibly a vassal or union or two in Europe) might be enough to earn the achievement. If not, then it's time to head down the west coast of Africa, around the Cape of Good Hope, and then back up East Africa. There are more countries there that can be turned into protectorates and, if necessary, it is also a staging post for trips to India where more protectorates can be acquired.
The Holy Roman method
If playing as the Holy Roman Emperor, the Imperial reform, Revoke the Privilegia will turn all Holy Roman Empire member states into vassals and decrease their liberty desire. Depending on how many member states remain when the reform is passed, this can be enough by itself to gain the achievement. This means that this achievement can be combined in a single campaign with the achievement, A Kaiser not just in name.
The brute force method
Another way to gain this achievement is to simply grow very big and then release 21 vassals or client states at the end of the game. As mentioned above, having this many vassals will completely stop the accumulation of diplomatic power, so it is best done at the very end of a campaign when all other goals have been achieved and no further play is desired. While it is not recommended to play a campaign in this manner with the sole goal of accomplishing this achievement, it can be used secondarily in campaigns played for other reasons. For instance, if a player had just completed the Over a Thousand! achievement, it would be a simple matter to release 21 subjects at the end of the game to also grab this achievement.