There are exotic weapons, but they're just refluffed versions of normal weapons.
5e has a philosophy of simplicity and a lack of exotic dependencies on feats and as such there is not an extensive list of unusual and exotic weapons. A double bladed axe is a greataxe with a fancy name. A dire flail is a flail. You can see how this is handled with the monk's Martial Arts feature:
Certain monasteries use specialized forms of the monk weapons. For
example, you might use a club that is two lengths of wood connected by
a short chain (called a nunchaku) or a sickle with a shorter,
straighter blade (called a kama). Whatever name you use for a monk
weapon, you can use the game statistics provided for the weapon in the
Weapons section.
The DMG likewise has some advice on exotic weapons, when describing Wuxia settings (DMG, p. 41):
Having players refer to a tetsubo or a katana rather than a greatclub
or a longsword can enhance the flavor of a wuxia campaign.
That said, there are homebrew versions designed to implement a category of "exotic weapons". Here is an example.