You forgot the part where Dracula rises from his grave and invades Massachusetts. Meeting up the the descendants of the Salem Witches!The population of that part of the US at that time is 200,000-250,000, divided equally between Pennsylvania and Ohio, with Ohio's population being around 1/3 Native at this time. Pittsburgh and Chillicothe are the largest towns at 1500 people or so each.
By contrast, Transylvania's population at the time would be around 2.5 million, roughly 55% Romanian, 30% Hungarian, 10% German, and 5% Roma. It's larger than the 5 largest US states, population-wise. A 1790 census reports 1.5 million but that figure doesn't include roughly 1/4 of Upper Hungary that was included in that map. Klausenburg/Koloszvar/Cluj-Napoca is probably the largest city, with 15,000.
I'm not certain the Ottomans would try anything in this era considering Transylvania is Habsburg territory and the Ottomans are distracted by Russia breathing down their neck. The Austrians would have virtually no trouble establishing authority in the region, considering they have about as many infantry soldiers as the Americans have people. However, Austria's attention is going to be on the Southern Netherlands, which are currently occupied by Napoleonic France. In OTL the Napoleonic Wars never really touched Transylvania proper, although fighting between Russia and the Ottomans happened in neighboring Moldavia and Wallachia.
I imagine that the most people in Transylvania won't want to leave their homes in North America, and there's no way to evict them, either, since there's so many of them. The Boyars and nobles would be hesitant to return to the Habsburg lands because they've had disagreements about the Habsburgs' respect for their autonomy and traditions. The American settlers in Europe would probably prefer being on the other side of the ocean, given anti-European sentiment was still fairly strong in that era.
The Transylvanian nobles would probably try to convince the Americans to grant them independence, since the feudal system practiced there is incompatible with US legal and societal mores and the people there are uniformly non-English speaking. The Americans also wouldn't be fans of the Hungarian nobles, since those are mainly Catholic. The US also doesn't have the military strength to take the territory, as it's bordered by mountains and full of castles.
As time goes on, I imagine both Romanian peasants and American settlers would emigrate to the US proper. Hungarian nobles would double down on their traditions and the US would have no choice but to acknowledge their sovereignty. Most of the Romanians would flee west into the Northwestern Territory, which the US government would probably encourage. Eventually, we'd have Indiana, Illinois, and Michigan having an overwhelmingly Romanian population, but by the 21st century most of them would probably be more or less assimilated into American culture with most speaking English and attending Orthodox churches. In Transylvania itself, I think that the presence of a large republic surrounding the territory it would cause uprisings against the feudal order, probably transitioning to a republican form of government in the mid-19th century. The US might try to reassert its claim on the territory after that, but if they manage to stay independent past 1900 they're likely to stay that way.
In Europe, I think the American portion would probably be settled heavily by German migrants who were financed by the Austrian crown to populate reconquered Ottoman lands. In time, the population of Americans and Germans would probably end up about even, or perhaps slightly favoring the Germans. I can't imagine the Austrian monarchy being any more tolerant of Native Americans than the US government, so there will probably still be tragedy there. As the region becomes embroiled in the conflicts and ethnic cleansings of the late 19th and early 20th century, it's difficult to know what will happen to the Americans as time goes on, but the region will probably not go to Romania as it did in OTL. It would probably assert itself as an independent republic if Austria collapses and then fall to whatever empire gobbles it up in the early 20th century, whether that's Communist Russia, Imperial Russia, or a Germany of whatever political inclination. In modern times it would probably be similar to the countries surrounding it, with the exception that it would be bilingual in German and English.
But in all seriousness though this has to be the most straight and informative answer to a low effort meme map I've ever seen. Absolutely Incredible!
One interesting thing is Ohio was made a state on March 1803 so depending on when this exactly happened Ohio could already be a state and Ohio will likely absorb the parts of Pennsylvania that came with it. The US just might have a State in Eastern Europe for a time at least. Austria could take over no problem but I don't see much of a reason for Austria to take over the region. Really depends on what Napoleon does and whether he'd back up the US should the region be invaded. Austria would probably let Transylvania be independent. Given how far away it is it'd be too difficult to properly control. The same could be said for Ohio and South West Pennsylvania but that's a huge congressional mess to let a state secede from the Union and it would give a few certain states some legitimacy when they try to seceded.
I'm curious to see how Transylvania would factor in the American Civil War. Would they be Neutral? Aid the North? Aid the South?
If they aided the South then I could see the North having no choice but to invade and occupy Transylvania. Holding a referendum vote after the war on US Statehood or Independence. Or maybe just Manifest Destiny the land of Dracula.