I have always been curious as to how how Romania retained such strong ties to its Latin roots, especially as it seems to be surrounded on all sides (except Hungary) by Slavic-dominant cultures.
Having read up a bit on Romania's early colonies, it seems they consider their cultural heritage to derive from Daco-Roman cohabitation. However, most other cultures I know which were conquered by the Romans seem to have deprioritized Latin identity at some point in favour of a more regionalized hybrid culture. See Gaul becoming France, Iberia becoming Spain, or the Byzantine empire eventually fracturing into Turkic, Greek, Slavic, etc. cultures. None of these seem to identify as strongly Latin cultures, although some may nod to their Latin roots.
Further, it seems that Romania launched several re-Latinization efforts in the 18th and 19th centuries, several hundred years after the fall of anything which could be construed as the Roman Empire. I'm somewhat familiar with their resistance to Slavic invasions, but what is it about this culture so far removed from Rome - both geographically and historically - which made them so enduringly Latin?
Bit late with an answer. I hope someone will read it. Haha
I will assume that by Romania, you mean the modern state and Romanian culture in the XIX century to the present.
The fact that Romanians spoke a Romance language was known since the late middle ages. Foreign travelers from the west highlighted this. It seems that the people themselves called one another "rumâni". Also, in the 1640's a Moldovan chronicler, Grigore Ureche, wrote in his book Letopisețul Țării Moldovei that us, Moldovans, trace ourselves from Rome: "de la Rîm ne tragem". While hinted, Latinism was not an important aspect of Romanian culture. Religion was the most important aspect.
The emphasis put on the Latin side of Romania comes during the age of nationalism and actually is a reaction to Hungarian nationalism. Hungarians started their national ideology before Romanians and a part of the narrative put emphasis on the fact that they were the ancestors of the mighty Huns and were the first who inhabited the Hungarian Plain and Transylvania. In the age of nationalism not only the right of the conqueror was important, but also the historical right. It was believed that those who came first in an area were the one who had a right to own a region. This is something usual in the Balkans and Eastern Europe where many people were part of an Empire. Now, in the XIX century, those people wanted independence.
The Hungarian claim ignited a reaction among Romanian intellectuals of the late XVIII and early XIX century. If the Hungarians claim Transylvania because of their historical right as descendants of the mighty Huns, than Romanians, the story goes, are more entitled to it because they are an even older people who descend from the might Romans. The Historical debate of who came first is still a hot topic even today.
The Romanian intellectuals in Transylvania were the first to make this into a coherent historical-linguistical theory and it spread on the other side of the Carpathians, into Moldova and Walachia. In short, Latin ancestry became a part of the Romanian national consciousness because they were great tools in the age of nationalism. It help lay claim to a distinct national feeling(the only Latin nation in the region), an important tool in the XIX century and for Independence(in the case of the Danubian Principalities) and for being granted more rights by the Habsburg and then Austro-Hungarian governments.
The emphasis on Latin genealogy was accompanied by a "latinization" of the language where many Slavic words were dropped and new Latin(or French, Italian) were adopted, the Latin script was also adopted. Also the name "Romania" was adopted and not "Dacia" for example. Romanian politicians also used the Latin card with France, the main Latin power of the era. This PR campaign helped, France(through Napoleon the III) was a great supporter of the Romanian cause, although financial and geopolitical aspects were also important. In Romania, France was seen as a Latin big sister and was the cultural model.
In short, the fact that Romanian was close to Latin and it was spoken in the 3 regions was known but the emphasis on everything Latin came in the age of nationalism.
The period of Dacian/Romanian history in which the Latin identity in the region endured is very poorly documented and subject to a large amount of academic debate. Many historians believe that the Daco-Roman identity was able to endure long after the Romans left the area but others believe that the Daco-Roman identity was completely swamped after the Roman army left Dacia and that modern Romanian identity can instead be traced to Romance-language speaking shepherds that migrated up from the Balkans long after the fall of the Roman Empire who are related to the Aromanian peoples who still live in the Balkans. Since we can't nail down for certain where modern Romanian identity comes from it is hard to tell why it endured so well.
Here is a chapter from an academic text that goes over the debate of where Romanian identity was formed, as you can see it's quite contentious so I can't tell for certain personally if Romanian identity was formed south or north of the Danube or if some kind of compromise is possible: https://books.openedition.org/ceup/935?lang=en
I cannot speak to the persistence of Latin identity in Romania, but I would take a strong objection to your suggestion that the modern French and Iberian identities are no longer Latin. The French identity draws as much from its Gallo-Roman ancient past as from its Frankish early medieval past, especially in language and religion. Most people in Portugal and Spain have names that originated in the Roman era (including Biblical names translated into Latin in ancient times) and the majority of Spanish and Portuguese speakers today live in Latin America.
Perhaps most significantly, all of these countries are overwhelmingly Roman Catholic, meaning that for centuries French, Spanish, and Portuguese people learned classical Latin in addition to their Latin-derived native languages, whereas Romania has spent its history as an Eastern Orthodox country in the Byzantine sphere.