They are familial possessions, sources of revenues and loyal soldiers.
This is only the case when the holder also has a higher title (such as the Prince of Wales). If you were the King or the heir, then you could hold both Northumberland and Norfolk or (as Henry II did) the Aquitaine and England ... but even he needed a lot of help to do it and was constantly fighting wars with his wife (the Duchess of Aquitaine) or his kids (Dukes of Aquitaine and Brittany).
Your example of Cornwall is non-traditional for that reason and didn't come about until the High Medieval era under Edward III. Prior to that, the Earls of Cornwall were generally also local lords in Cornwall, with one major exception being the super-rich Richard of Cornwall who also happened to be King on the continent. The HRE is also not a great example because the politics of the Empire were very unique to the Empire and also greatly subject to the Pope's influence. The HRE was at the same time both less local (since everybody was nominally part of this huge Empire) but also more local (as the politics were dominated by in-fighting between the different cultural and political blocs).
I think you have it backwards: they became less cultural toward the end of the Middle Ages as duke-doms were created willy-nilly, particularly in the Renaissance, to reward sycophants. If you made a list of all of the Duchies in Western Europe circa 1289, you would have a pretty good list of the various cultural groups. That's not to say that they weren't also familial possessions, but most Duchies changed families at least a few times during the Middle ages, but the character of the Duchy itself would not necessarily change with it. France is a great example of this: you've got Normandy, Brittany, and Anjou all next to each other, and you'd have a hard time finding three less similar cultures all kinda-sorta called "French." This is still true today, though a lot less so, though the Bretons are still pretty distinct. They're a classic example because they were the only hold-outs in all Western Europe to never have been conquered by barbarians -- so "Breton," or "Romano-British" was far more a cultural and political construct than a familial one. Bretagne was largely independent of France for much of the middle ages despite not being much of an economic or military power. The people were just
different, and if you were the Duke of Normandy or even the King of France you weren't too thrilled at the prospect of putting up with tons of unrest by occupying Brittany, particularly because you already had all the nice champagne and bordeaux.
Part of this topic is confused because a BM "Duchy" is less like a historical duchy and more like a CK2 duchy -- it's the political entity bigger than a county but smaller than a kingdom. In BM the local culture does not impact us very much and even a totally opposite religion can be purged by a new lord in a fairly short time span. So our duchies are not bound the same way and can be re-drawn a lot more easily than a medieval duchy. Shoot, a lot of those medieval borders still hold, at least culturally: the Aquitaine has always been a different flavor of French from Anjou or Champagne and York and Cornwall or York have always been very different from London (even if the Duke of Cornwall is the heir - ask any Cornishman whether he thinks the Duke of Cornwall has got the first thing to do with Cornwall and he'll tell you no, assuming you can understand his accent).
