Interestingly, this word "viineri" comes from "Viennoiserie" which is the French phrase for "things of Vienna" - Essentially croissants, pain au chocolat, brioche... So the food we think of as French originated in Austria but through Finnish we call it Danish! There's a little European cultural-culinary-linguistic tour for you!
It's much more likely from "Wiener", which means "of Vienna" in the Scandinavian languages. Spell it in Finnish and put an 'i' at the end and you got "viineri". Exact same as "kioski", "pankki", "posti" etc. These pastries originated in central Europe and spread directly to nearby countries like France, but to northern Europe it spread through Denmark. A Viennese baker made it wildly popular in Copenhagen, where they called it "Wienerbrød" "Bread of Vienna". It quickly spread to the rest of Denmark and northern Europe. In some languages the Danish name "Wiener(brød)" was used, while others like English and German called it "Danish pastry" and "Kopenhagener gebäck".
'Danish' denotes nationality or country of origin. As it isn't known with certainty that the pastries were made in Denmark, or that the recipe comes from that country, referring to the country does not appear to be accurate or grammatically correct. Bacon from Denmark is usually referred to as bacon rather than Danish bacon. As the items are pastries, perhaps using this in translation would be a simpler and clearer alternative.