Upvote:2
If I understand the law correctly, Kosovan identity cards are only issued to the citizens of the Republic of Kosovo:
... për shtetasit e Republikës së Kosovës.
This means that you will need Kosovan citizenship (and hence lose your Danish citizenship) to apply for a Kosovan identity card.
Be aware though, that Denmark has already enacted modifications to its nationality law, which will take effect on September 1st 2015, after which you can keep your Danish citizenship even if you are granted another citizenship.
Upvote:2
Usually it works the other way around: You need to be a citizen to get an ID-card or passport (some countries do issue ID documents or passport-like travel documents to non-nationals in various cases but those typically have another name). The card is just a convenient way to prove it but citizenship is legally defined based on the circumstances of your birth and other factors and does not depend on actually holding a specific document.
Consequently, if you are eligible for an ID card in Kosovo it would seem to imply that you already are a Kosovo citizen and could therefore already have lost (or be at risk of losing, e.g. by court order) your Danish citizenship, whether you do get a card or not. Of course, if that's the case, actually getting the card increases the likelihood of being found out by the Danish authorities in one way or another. But it would not change your legal status.
Also, I don't know Danish nationality law but whether you were born in Denmark, whether you acquired the Kosovo nationality after becoming Danish, failed to renounce it upon naturalization or whether it is possible to renounce it at all could all make a difference. You need to consider all this to precisely assess your situation. Wikipedia also suggests Danish nationality law has just become a lot more accommodating with dual citizenship so being considered a citizen in Kosovo might not be as much of a problem as you think.
One further complication in this case is that Kosovo isn't a fully recognized state internationally but I don't know whether this would make a difference, especially considering the fact that it has been recognized by Denmark.