The precise legal and political status of the Spanish possessions in the Americas is a bit uncertain to me.
I have sometimes read mentions of various regions as kingdoms.
For example, the region in northern South America called the Viceroyalty of New Grenada has bee n called by other names.
The Viceroyalty of New Granada (Spanish: Virreinato de Nueva Granada [birejˈnato ðe ˈnweβa ɣɾaˈnaða]) also called Viceroyalty of the New Kingdom of Granada or Viceroyalty of Santafé was the name given on 27 May 1717,1 to the jurisdiction of the Spanish Empire in northern South America, corresponding to modern Colombia, Ecuador, Panama and Venezuela.
https://en.wikipedia.org/wiki/Viceroyalty_of_New_Granada
Furthermore, the region was called the New Kingdom of Granada for about two centuries before 1717.
Two centuries after the establishment of the New Kingdom of Granada in the 16th century, whose governor was dependent upon the Viceroy of Peru at Lima, and an audiencia at Santa Fé de Bogotá (today capital of the republic of Colombia), the slowness of communications between the two capitals led to the creation of an independent Viceroyalty of New Granada in 1717 (and its reestablishment in 1739 after a short interruption).
https://en.wikipedia.org/wiki/Viceroyalty_of_New_Granada#Colonial_history
The monarchs of the Spanish kingdoms had a long full title. From 1516 the list of kingdoms they ruled or claimed ended with "the Indias, the Islands and Mainland of the Ocean Sea" their title as ruler of the American possessions.
In 1581 that title was changed to king of "the East and West Indias, the Islands and Mainland of the Ocean Sea".
From 1808 to 1813 Joseph Bonaparte used the title of "King of the Spains, the Indias".
In 1813 King Ferdinand revived the old title including "the East and West Indias, the Islands and Mainland of the Ocean Sea".
http://eurulers.altervista.org/spain.html
So the Spanish monarchs always used one royal title for all of their overseas possessions. They never used a title like King of "New Spain, New Granda, Peru, Rio de la plata" that would imply they were separate kingdoms.
And if various regions of Spanish America were established as official kingdoms, why wouldn't the king of Spain change his title to list those kingdoms separately? Why would any ruler leave some of the kingdoms that they ruled out of their title?
Someone whose full title was:
King of Castile, Leon, Aragon, both Sicilies, Jerusalem, Navarra, Granada, Toledo, Valencia, Galicia, Majorca, Sevilla, Sardinia, Cordova, Corsica, Murcia, Minorca, Jaen, the Algarves, Algeciras, Gibraltar, the Canary Islands, East & West Indias, the Islands & Mainland of the Ocean sea;
Archduke of Austria;
Duke of Burgundy, Brabant, Milan;
Count of Habsburg, Flanders, Tyrol, Barcelona;
Lord of Biscay, Molina;
wouldn't object to adding a few more kingdoms to the list.
On the third hand, the kings of Spain ruled a number of separate European kingdoms outside of Spain. Those were personal unions, kingdoms which had the same monarch as the Spanish kingdoms. They included the Kingdom of Sardinia, the Kingdom of Sicily, and the other Kingdom of Siciy, usually called the Kingdom of Naples. Those kingdoms were administered by viceroys.
here is a link to a list of viceroys of Sardinia:
https://en.wikipedia.org/wiki/List_of_viceroys_of_Sardinia
And a list of viceroys of Sicily:
https://en.wikipedia.org/wiki/List_of_viceroys_of_Sicily
And a list of viceroys of the other Sicily:
https://en.wikipedia.org/wiki/List_of_viceroys_of_Naples
If the status of the lands ruled by Spanish viceroys in the New World was the same as the status of lands ruled by Spanish viceroys in the Old World, the Viceroyalties of New Spain, New Granada, Peru, and the Rio de la Plata would be separate kingdoms in personal union with the kingdoms of Spain.
Portuguese Brazil was a kingdom from 1815 to 1822 when it declare dindepence from Portugal.
If the viceroyalties of New Spain, New Granada, Peru, and the Rio de la Plata were separate kingdoms, then along with Brazil they would have been the five largest kingdoms ever ruled by Europeans.
And If the viceroyalties of New Spain, New Granada, Peru, and the Rio de la Plata were parts of the Kingdom of the Indias, the Islands and Mainland in the Ocean Sea, that kingdom would have been by far the largest kingdom ever.
And now the OP says they have found mentions of the Kingdom of New Mexico among the Spanish possessions. So that seems to indicate that the Spanish did not limit their use of the word kingdom to the four viceroyalties, but also used it for smaller regions.
So that is an interesting question. How many regions of Spanish America were called kingdoms, and how usual was it to call them kingdoms, and what was the meaning of calling them kingdoms?
