"Why the Finnish language?"
The sentence doesn't really stand on its own, so it can be a bit confusing lacking the context. However, you have a solid point.
Consider these two examples:
|Minä opettelen suomea.||I am learning Finnish.|
|Miksi suomea?/Miksi suomen kieltä?||Why Finnish?/Why the Finnish language?|
|Koska se on niin kaunis kieli.||Because it's such a beautiful language.|
|Minusta kaikkein vaikein kieli on suomi.||I think that the most difficult language is Finnish.|
|Miksi suomi?/Miksi suomen kieli?||Why Finnish?/Why the Finnish language?|
|Koska se on niin erilainen.||Because it is so different..|
So you can see that the same sentence could be translated with the partitive, given a different context.
The question does not stand to the expected answer. "Miksi suomi?" is kind of correct but "Miksi suomen kieli?" I try answer what should be wrong answer "Miksi Suomi" and still pass as good, probably cause answers are not case sensitive, so that males that question not just confusing but also useless.
When suomi, ruotsi, or some other language is in the nominative case, it can't directly modify the noun kieli. But it can be put into the genitive case, which ends in -n.
So either 'Miksi suomi?' or 'Miksi suomen kieli?' Either 'Miksi ruotsi?' or 'Miksi ruotsin kieli?'
When there's the verb on, that verb can link nominative suomi to kieli. Thus 'Suomi on kieli', "Finnish is a language".
I believe the verb on can't link suomi to a nominative adjective, since the language suomi seems to be an uncountable noun. So the nominative adjective needs to precede kieli, which is a countable noun.
Thus 'Suomi on kaunis kieli', "Finnish is a beautiful language". Though 'Suomen kieli on kaunis', "The Finnish language is beautiful", is also a grammatical sentence since there on is linking the adjective to kieli.
So "Why the Finnish Language?" needs to be "Miksi suomi?" Which makes sence. But then "What is the hungarian language like?" needs to be "Millainen unkari kieli on?"
People in the comments for that one say you need kieli for context despite unkari being in lower case. Now I come back to this one, and it's the other way around again.
Am I missing something or is Duolingo just being that picky?
Here, both suomi and suomen kieli should be accepted. But since the noun suomi is directly modifying kieli in the second option, it needs to be in the genitive case, suomen.
In writing, it's clear that 'Miksi suomi?' is asking about the language. In speech, you'd need additional context to know if that question was about the language or the country.
As for "What is the Hungarian language like?", I also wanted to translate that to 'Millainen unkari on?' However, it seems that while the country Unkari is treated like a countable noun, the language unkari is an uncountable noun.
When an adjective is linked to a singular countable noun by the verb on, that adjective is nominative. But when an adjective is linked to an uncountable noun by on, then that adjective is partitive.
Following those rules: 'Millainen Unkari on?' "What is Hungary like?" But: 'Millaista unkari on?' "What is Hungarian like?"
But kieli is a countable noun, so if the adjective millainen modifies kieli, then millainen needs to be nominative.
Thus 'Millainen kieli unkari on?' "What kind of language is Hungarian?" Here, millainen directly modifies kieli, and unkari is linked to the noun phrase millainen kieli by the verb on.
There's also the possibility of 'Millainen unkarin kieli on?' "What is Hungarian/Hungary's language like?" Here, it's genitive unkarin which is directly modifying kieli, and the noun phrase unkarin kieli is linked to millainen by on.