"De forente meksikanske stater"
Translation:The United Mexican States
Yes, it should by modern language standards, but no, that's not how the official name of the country is written.
You'll see many older names lacking the definite suffix, and this is a result of the strong Danish influence on the Norwegian language at the time. Since they're proper nouns, they're seldom changed to conform with current language norms.
In most settings, the country will be referred to as "Mexico", plain and simple.
It's confusing to me that in Norwegian the country spelled two different ways-- "Mexico" vs. "meksikanske". It seems to me the former should be "Meksiko"! Is there a reason for this?
Yes, it's confusing, but yes, there's a reason for it.
The "x" in "Mexico" is preserved because it's a proper noun, and we try not to mess too much with proper nouns - just like you wouldn't change the the spelling of someone's name based on your own pronunciation. "Meksikansk" is just an adjective, and we have less qualms with changing those to conform with Norwegian orthography.
You'll see the same thing with "Canada" and "kanadisk". Curiously, both "kanadisk" and "meksikansk" still have optional spelling variations preserving the "c" and "x" respectively - but these forms are much less used than the ones I've listed.