the soul of Azem

[Gemina]

There's nothing in the story that actually binds us to Azem's identity anyway. It has no tangible impact on anything except Hades' sentimentality and our gaining command of the summoning spell from the crystal.

Gra'ha does have speculations that certain characteristics are also transferable. Hermes' many incarnations all exhibited an affinity with the celestial, instability, and psychosis. Souls also have hues/colors that are recognizable by certain aether sensitive individuals like Hythlodeaus, Hades, and to a much lesser extent, Y'shtola. In the case of Azem, it's not just the color of their soul and appearance (if playing a Hyur) that is recognizable by Emet, but also the behaviour. Sure, Emet is a sentimental guy underneath his cold exterior, and overlapping behavioral traits does very little to link incarnations of a soul; but this anecdotal evidence still carries a lot of weight due to the additional factors involved.

Even though there is no concrete proof that the soul possesses distinctive traits, it's also one of those things that there is also nothing to prove that it isn't true. I think it is better and more accurate to say that memories is what separates the incarnations of a soul from each other. However, this alone is enough to give each lifespan their own identity.