As in why do dragons in European and Chinese mythology seem very similar to each other? At least why do they share similar characteristics.
They are actually not the same creature. (For a lot of Chinese culture the English-speaking world often come up with a familar item that's analogous to the Chinese item and add the word "Chinese" in front of it instead of calling it by its Chinese name.) Here's a more in-depth answer from previous thread.
There's this book called "An instinct for Dragons" by David E. Jones that presents a theory that I like, personally. It argues that dragons are a combination of the things that apes are genetically programmed to fear.
Dragons are scaley and long, like snakes. It is actually theorized that the human brain passively looks for snakes all the time, since studies have suggested that we seem to notice them much more easily than other creatures. But anyway, dragons have the obvious snake characteristics. Second up are the great cats. We come from Africa, and lions and leopards and their ancestors would have posed a large evolutionary threat to apes and the early hominds. The dragon typically has a mouth much like a great cat, with long, sharp canines. Predatory birds, which would have threatened young apes, give the dragon its flight. The innate fear of these creatures may explain why dragons pop up in separate mythologies; they combine characteristics of the primary threats we faced.
Part 1:
As cecikierk mentioned while we colloquially use the English word "dragon" in both cases they should be considered different mythological creatures and it's more poor translation that leads to this question.
Chinese Dragons
The "Chinese Dragon" in it's earliest references is primarily referenced as having an almost serpent like figure. The Chinese word for dragon (龍) began it's life in oracle bone script as a picture of a whiskered serpant (Wiki pics/etymology). We're not 100% sure where dragon's entered into Chinese culture but they have a lot of overlap with symbolism of a serpent coil. We've found at least one statue that we believe may be a dragon statue from the Yangshao culture (circa 5000 bc), and jade military rank crests that have the coiled dragon pattern from the Hongshan culture (4500 bc) but around 300 BC we're positive that the concept of a dragon exists as historian Chang Qu writes about the unearthing of "dragon bones".
We know now that many ancient Chinese culture's unearthed dinosaur bones of a species now known as [Mei Long](https://en.wikipedia.org/wiki/Mei_(dinosaur) (or "sleeping dragon") and many of these fossils have been discovered in this "coiled" sleeping pattern Picture. I don't know enough about archeology to know why so many were discovered in that formation but it's clear that this had a huge impact on the dragon concept and symbolism around China.
As we know when myths and stories evolve and get passed down through generations they morph a great deal. Things get added, things get removed and somewhere between the period of 300 BC and the life of Han Dynasty scholar "Wang Fu" (82 AD - 167 AD), Dragon's became less dinosaur like and much more like sea serpents. Example of early Han period sculpture which shows the more long form Dragon.
In Fu's writing we discover that there is the long standing myth that the ancient Chinese people believed themselves to be descended from Dragon's, which were seen as a sort of common ancestor between humans and "common animals". This lead to the common description of the "long dragon" which had, according to Fu, 9 anatomical similarities:
Antlers of a stag,
head of a camel,
eyes of a demon,
neck of a snake,
belly of a clam,
scales of a carp,
claws of an eagle,
soles of a tiger,
ears of a cow.
It's likely that the suggestion of a Dragon being a common ancestor slowly morphed the image of the Dragon further and further to show all the creatures represented within.
It's also worth noting that Chinese Dragon's do not have wings (at least in most depictions). The source of their flight, and ability to control water isn't a physical trait, but rather a mystical energy which stems from their "Chinu" 尺木 or a lump on their head.
European Dragons - The History
European Dragon's on the other hand seems to primarily exist due to extended legend rather than any particular discoveries. To understand the British Dragon (what you may know as a "Fire-Breathing Dragon") you need to understand a bit about other European dragon myths as unlike the Asian dragon the British one didn't stem from fossils. For while Greek, Roman, Chinese and Persian scholars had brief flings with archeology, Europe in the Middle Ages is rather a barren wasteland in this field. The few references that we have to fossil's at this time suggest they were thought of as "works of the devil" or remains of "vis plastica" a molding force that God used to give shape to the world.
From what I've turned up, a lot of the European belief in dragon's (specifically the British Fire-breathing dragon) is primarily influenced from other mythical cultures (Greek, Germanic, Norse, and even Roman fables).
Most of these seem to stem from the classic Greek δράκων (drákōn) which while little is written about, seems to be very serpent like in Greek classic art Example of a Greek Dragon on a Vase. Which is remarkably similar to the Chinese version of a Dragon that would have been evolving in concept at roughly the same time. One of the most noted examples of the Greek Dragon would be Λάδων or "Ladon", a serpent dragon slain by Heracles, which we hear about in Hesiod's "Theogony".
Throughout the Germanic/Nordic regions of the middle ages the overlap between dragons and serpents continues. In Germanic mythology the "wurm/wyrm" is a type of serpent (often pictured with two legs and wings), in Finnish "lohikäärme" is believed to either stem from louhi (Mountain) käärme (Snake) or the Norse word lógi meaning "fire".
For those less familiar with how all these societies inter-mingled in this time period a short (and way too over simplified version) is that Britain in the Middle Ages was a hodge-podge of mainland cultures. There were Roman, Germanic, Saxon and Scandinavian presences here that brought with them myths, beliefs and language from the Greeks, Romans, Norse, and Old Germanic cultures, as well as possibly passing along myths and stories from further reaching cultures and experiences from the Middle East, Northern Africa and into Asia (it's worth noting that while this is commonly listed as speculation as it is hard for us to validate the passing of verbal myths, it's pretty viable that culture was easily spread across "The Roman Empire" which spanned from modern day Egypt, Sudan, Syria & Turkey all the way along to Spain and up to the tip of Britain Map of Roman Empire 450)
When we take this into account it is likely that old Norse and Greek legends that traveled to the UK in the Middle Ages crafted how dragons were viewed there. Something that may help validate this is that in both Norse/Germanic and early British cultures Dragons were seen as frequently guarding treasures. They were almost obstacles to be defeated and overcome on a quest, whereas Asian Dragon's were more benevolent creatures - almost a mythical extension of the universe and life aimed to do good and bring balance.
How The British Dragon Came to Be
In the west the most common view of a "Dragon" is what may be called more formally the "English Dragon" or the "British Dragon" (yeah I find it really weird that there is a more formal title for dragons..). The primary distinction here is that somewhere the British Dragon veered away from the European and Greek serpents and started to evolve an imagery of its own.
There are likely a few things that influenced this, but these are much more speculative than research I've seen about Chinese Dragons:
This is also likely the reason why Dragon's were seen as terrible beasts that needed to be defeated. The fire-breathing was tacked on as a symbol of the devil and hellfire, and they started stealing Princesses (as in the Middle Ages royalty were seen as those chosen by God, and an extension of God on earth. Therefore a beast of Satan would want to displace their influence by placing them in a tower or what-not).
I think people imagine some now-extinct real dragon-like thing. But a hypothetical animal isn't needed. If you've ever hung out in a swamp where there are alligators, snakes, and vultures, you'd see how easy it would be to come up with a dragon.
