Scandinavia is often mentioned by people campaigning for Scottish independence (more frequently than Ireland, which really is a bit odd).
However, most Scots don't actually know that much about Scandinavia, so let me try to describe what it means to be Scandinavian.
Let me first point out that Scandinavia is really just Denmark, Norway and Sweden. The wider group of countries that also includes Finland, the Faeroe Islands and Iceland (and sometimes also Greenland) is called the Nordic Countries.
The main bond uniting Scandinavia is the fact that the languages -- Danish, Norwegian (Bokmål & Nynorsk) and Swedish -- are mutually intelligible after a bit of exposure. Danish and Swedish are probably about as different as English and Scots (and to stretch the analogy a bit further, Norwegian is then like English spoken with a Scottish accent). There's obviously also quite a lot of shared history, but a lot of it involves wars between Denmark and Sweden.
All Danes, Norwegians and Swedes will agree that they are indeed Scandinavians, but it's not an identity that can be separated from the actual nationality. If you're Danish, you're by definition also Scandinavian, and you can't be Scandinavian without also being Danish, Norwegian or Swedish.
The Nordic Countries (not just Scandinavia) have operated a passport union (which allows all Nordic citizens to travel freely without a passport) since 1954. It's quite similar to the Common Travel Area uniting the UK and Ireland. The Nordic countries have not traditionally allowed dual citizenship, but instead it's somewhat easier to become a citizen in one of the countries if you were born in another Nordic country than if you were born elsewhere.
There used to be a currency union, but it broke up in 1914. This is the reason all the countries call their currencies the crown (krone/krona), although they aren't pegged to each other any more.
Otherwise, there aren't many tangible benefits to being Scandinavian. There have been several attempts at creating some kind of political union in the past, but these have typically failed because the individual countries don't actually agree on very much. Also, Denmark typically is keen to include Iceland in everything, and Sweden doesn't want to exclude Finland, which means that all Scandinavian projects end up involving all the Nordic countries.
However, something still unites Scandinavians. It's very clear if you go to an international meeting (such as an academic conference): The Danes, the Swedes and the Norwegians will typically end up as one group in the evening because they share so many linguistic and cultural bonds. It's just something you don't think much about until you leave Scandinavia.