It does. However people often assume that "non-Western" = "crazy and lazy Muslim" in debates
We have our share of Alevis (which are kind of Muslim), Lebanese, Syrian, Palestinian, Iraqi, Iranian, Ethiopian and Eritrean Christians. Also a bunch of Ezidis (Yazidis) who have scrambled to support the Ezidis in Iraq.
I also know several Atheist Danish-Palestinians (Yahya Hassan is one) and "Agnostic Muslim"-ish Danish Afghans and Kurds. Lots of Kurds. As it read in a paper zeca posted (thanks, zeca): "Compared to the infidel, the Kurd is Muslim"
They mostly are pretty lax on Islam however there are "honour culture" issues among some. There have been incidents of honour killings among Danish Kurds (you UK'ers might remember "Banaz - a love story" - she was also of Kurdish heritage).
We also have some Armenians around which are also labeled "non-Western".
Same with our large group of Bosnians, Kosovars and Albanians (and Serbs and Croats (who magically became "western" in 2013) and Montenegrins and Macedonians). They are also counted as "non-Western" despite they have in general been very good at adopting their new host country. Minus the ones that ended up in the gangs. Quite a bit of those are Romani (as with the Romanians) but those that settle and work don't mention that because they don't want the associated stigmatisation. There are about 15,000 settled Romanis that came from the former Yugoslavia in the 70's-90's according to a Danish-Romani cop (!) I spoke to once and estimated about 100,000 Danes with some kind of Romani background accumulated through the last couple of centuries.
For Denmark you are "non-Western" if you are not from the EU, Iceland, Norway, Australia, New Zealand, some of the European mini-states or North America minus Mexico