The referenced older answers by u/ancienthistory and u/itsallfolklore provide excellent overviews of some of the methodological pitfalls of comparative monster studies. I'd just like to point out that the difficulties of translating terms for mythical creatures aren't purely a problem of modern culture/academia. Since these entities don't have the same kind of obvious physical referentiality as, say, a horse, interpreters have to fall back on other points of linkage.

The decision by Hellenistic translators to generally render Hebrew & Aramaic tannīn as Greek drákōn, for instance, may be connected to the narrative role these beings play in particular texts, as adversaries whose overthrow by a hero or divine figure points to the latter's righteousness (the Book of Daniel is a good illustration here). This is in spite of the quite different etymologies of these terms--tannīn probably derives from a verb meaning "to twist, stretch, coil [like smoke?]" while drákōn seems to be from the verb "to see," perhaps referencing the hypnotizing eyes of the monster.

Much later, Islamic scholars used the Arabic tinnīn (a close cognate of tannīn) as an equivalent for the New Persian azhdahā, though the latter comes from an Avestan term meaning something like "snake-man" or "barbarian-serpent." This connection seems to come from a perception that both were giant, serpentine creatures, even though (despite what you may read!) the azhdahā fulfills a pretty distinct cultural function in New Persian literature.

Turning to the Chinese "dragon," there's a fascinating attestation of the diverse ways this creature was understood in premodern cultural spheres beyond East Asia. The Rasulid Hexaglot is a Yemeni text from the 14th century CE. As the title indicates, it indexes six languages--Arabic, Persian, a Turkic dialect, colloquial Greek, Armenian, and Mongol--in an extensive series of word-tables, grouped by subject matter. Not all entries include all six languages, but it's still a fascinating insight into the art of medieval translation.

In a segment on animals, there's an entry that runs: [Arabic] al-timsāḥ [Persian] nahang [Turkic] ılan balıq [Mongol] lū. Al-timsāḥ is pretty straightforward; it means "crocodile." Though medieval dictionaries often use nahang to translate al-timsāḥ, it really has a broader meaning referring to dangerous aquatic creatures/monsters that drag their victims down into the depths--sometimes crocodiles but also sharks, hippos, and more mythical beasties. In Modern Persian it actually means "whale." Ilan balıq literally means "snake fish"; in Modern Turkish this seems to refer to eels, though I'm not aware of any broad medieval usage (I'm not by any means a Turkic specialist, though!) But lū (from what I understand--again, not a Mongolianist) is again pretty straightforward; it's a borrowing from the Chinese lóng, the creature that Modern English usually calls the Chinese "dragon."

What seems to be going on here is an attempt to translate the years of the Chinese calendar. An adjacent entry translates Arabic "the sheep" into other languages as "year of the sheep" (generally "Year of the Goat" in English). And, as it turns out, Modern Persian continues to use "nahang" for the Chinese calendar's "Year of the Dragon," even though Modern Persian doesn't make much if any other association between nahang (whales) and dragons (for which the standard translation is the term I give above, azhdahā.)

All of which is just to illustrate that there is a millennia-long interest in trying to equate mythical beasts from different cultures, and that the specific equations made between, say, the lóng and the dragon, are highly dependent on specific contexts.

Bad translations that shoehorn entirely different things under the same label. More can always be said on the matter, so if anyone would like to address the (mis)application of the term 'dragon', please feel free to do so! For the meantime, OP, here's u/itsallfolklore addressing the ostensible commonality of the dragon.