The Echo is seemingly awakened by a reincarnated Ancient soul witnessing something that awakens echoes of their past lives, often via traumatic events like a Calamity, or a meteor shower causing a flashback to the Final Days. Going by current evidence, it would seem that souls created post-Sundering can't gain the Echo (leaving aside the Resonant), but it hasn't really come up.

Dragons are effectively unsundered, since, well, they weren't here for the Sundering. However, they live on a much slower time scale than the Ancients did. Theoretically, the only thing stopping a wyrmling like Ohl Dee from becoming a nigh-godlike being like Midgardsormr is time, but it's a lot of time. That said, a dragon couldn't possess the Echo because there's nothing for them to be an echo of, unless somehow an Ancient soul fragment reincarnated into a dragon.

The only other confirmed aliens we've met so far are Gilgamesh, Omega, and the YoRHa androids. Gilgamesh is hard to call, but Omega and the androids, being machines, don't really qualify at all, what with not having souls. (And the Ea in Ultima Thule, but as they're nothing but aetherial souls......)