In the American and Canadian English dialects I grew up speaking, I have heard "I am trying a glass (or bottle) of wine," though never "I try a wine." When a specific brand or type of wine is referenced, the sentence structure is closer to that of Danish (e.g., "I try/am trying a Chardonnay"), but I am unable to think of instances in which "a wine" would be used in this manner. Strangely, hearing "a beer" rather than "a wine" in this context would not sound incorrect to me.
I think the correct idiomatic translation into American English would be "I taste/am tasting a wine", in the sense of "to test" or "to sample", for example, to decide if I want to buy it, to compare it to other wines, etc. If I wanted to say I taste (that is, detect the flavor of) wine in something, then "at smage" might be the appropriate Danish verb.