The classical dfinition is the geographical peninsula which does not include denmark. The concept of the country grouping scandinavia is a marketing thing, at it shows up as sweden starts tryign to belong to western europe post the world wars. Culturally of course (aside from lingually) we're much closer related to eastern europe, just look at the culinary culture (a good indication because there is less cultural interventionism in culinary culture than in lingual).
Well, as the Kalmar Union signifies, a common "Scandinavia" was a thing at the beginning of the game. The country grouping seems to have an undeniable historical basis and afaik the beginning of the unity marketing was a real thing borne by art and culture at the start of the 1800's with the Kalmar Union as the template. Culturally, Denmark (and by proxy Iceland, Norway and the semiautonomous nations) and Sweden (and by proxy Finland and the semiautonomous nation(s)) are related for good or bad both historically, culturally and politically. Now, tsariate Russia and the baltics are related to Scandinavia and particularly Estonia for historical reasons, but the relation there has always been of a far less political dimension.
As for Copenhagen being bossy, I think most people in Denmark outside of Copenhagen would agree. We have a discussion about "the rotten banana", which is basically anything outside a 50 km radius of Copenhagen and it is borne by a massive centralisation of power away from regions and communities towards the national assembly and beyond since the 1990's, but that is off topic.