Yes it does. The exeption is Finnish which is quite remote from the other Scandinavian languages (it's only related language in Europe is Hungarian for some obscure reason). But Norwegian, Swedish, Danish and Icelandian people understand each other completely. Of course we also share a common culture in most other respects.
I can't give a very good reason for not having a single name on these languages, but obviously having one of your own is an important part of a country's identity. Furthermore there has never (yet) been a Scandinavian empire and commonwealth, with a common administrative system and so on.
There has however been several types of Scandinavian unions. The rather loose union between Sweden and Norway ceased in 1905.