There are several aspects of the Scandinavian situation that were noticeably different from the German one in ways that would make a 'German fashion' unification difficult.
1. The number of states. In Scandinavia the only two that mattered were Denmark and Sweden, Norway was still closer to a mere territory they fought over. Not even a little brother. Meanwhile Germany was filled with little statelets, smaller states, bigger states, etc. If you put all the German rulers into a room, you had dozens or hundreds of people. For Scandinavia, you had the King of Denmark and the King of Sweden. The former might seem more troublesome, but it becomes less a contest of ego and personality when there's a crowd. It allows parties to form, compromises, and majorities. For Scandinavia in the 19th century, you'd just end up with two kings butting heads. Now that doesn't disqualify an agreement being made, but it does require certain personalities being king at the time. Alongside the proper party politics in the riksradet that would choose pan-Scandinavianism over Danish or Swedish nationalism.
2. Germany was right in the center of Europe. There's a reason much of pan-Germanism is said to have started during Napoleon's occupation of Germany. Germany was occupied, its people conscripted to fight for foreign rulers, had armies marching over it constantly, etc. With Poland partitioned, the Russian bear was right to their east. France was still to their west. The situation was either that Germany became a Major military power able to defend itself from all sides, or it would be the battleground of Europe (as it arguably had been for centuries). Scandinavia is not in such a position. Its main such area is the Sound, but they no longer had the strength to defend even that from Britain and Russia. Germany had the advantage (if a really bad one) of being clearly shown that it was in its best interest to properly unite, or it would never be able to stand against its neighbors. Not quite sure how you could get Scandinavia to receive such a clear message. Even a British-Russian war would likely have other theaters it could use as proxy battlegrounds. Maybe during the Napoleonic War, Britain just doesn't Copenhagen the Danish Fleet but occupies Zealand. Maybe even a bold move forces Norway, already dealing with famine and poverty from the the naval blockade of Britain, to surrender. With it appearing all of Denmark-Norway besides Jutland was thus occupied or surrendered, Napoleon might allow Russia to not merely take Finland but Sweden proper. If the Russians were looting Stockholm like Napoleon looted Berlin and conscripting Swedes to fight in their armies while Britain was press-ganging Danes and Norwegians into serving their ships, you could see greater pan-Scandinavian sympathies afterwards. Less 'they are our natural ethnic and linguistic brothers' and more 'they're our brothers and comrades in our fight to not be used and abused by our more powerful neighbors'.
So while I won't say it would be impossible, there would be several caveats to have a German style unification of Scandinavia. While it might be possible to accomplish it in the mid-1800's through changes in the Scandinavianism movement, Frederick VII's succession, and/or the Second Schleswig War, you might be stretching it a bit. You'd probably have to go all the way to the Napoleonic Wars. Make them even more devastating for the Scandinavian countries, although the OTL Napoleonic Wars were already harsh on them, so that both Sweden and Denmark are knocked out of any historic pride about their power and shown they can't stand alone. I'd also say Norway needs to become a proper kingdom. Three kingdoms is better in that it allows 2-1 votes, preventing a Danish-Swedish union from being paralyzed by the stalemate. Potentially a Norway that surrendered to Britain in the Napoleonic War could receive a boon in an alt-Vienna. A Danish prince marries an English princess as a compromise for Norway's ruler. Then a Swedish marriage a generation later solves that. Although it would be unlikely a Scandinavian unification would end up with a single 'higher' ruler like Prussia's king becoming Emperor, unless you end up with a Frederick VII scenario where one king has no heirs and his rule might fall to another Scandinavian king. It would be more likely, in my view, that a Scandinavian union would be merely involve a federal parliament for pan-Scandinavian affairs while the royal families stay constitutional rulers of their respective kingdoms. Then enough success in an alt-Schleswig War where the three cooperate against the weaker Greater Power of Prussia could restore pride and reaffirm that such a union was for the best. Denmark keeps Schleswig, although probably not Holstein. They then look east to 'liberate' Finland from Russia, thus becoming a minor power as time passes and an alt-WWI approaches.