So we know that the Dragonborn absorbs dragon souls whenever he kills one. But what exactly happens to the dragon soul after that?

Does the consciousness of the Dragon get assimilated into the Dragonborn? If so, does (or if not, should) this have an impact on his soul as well?

Basically, is there any precedent in the lore as to what happens when you absorb another beings soul? Just curious.

As far as I know, this is the first time we've heard of Dragonborn's absorbing Dragon soul's, but Dragon's haven't been around in so long there's really nothing about it. Not only that, but the only Dragonbrorn we know of for sure is Tiber Septim (and supposedly his line) and he/they didn't slay any dragons. The only thing I can think of in terms of precedent is when it comes to soul gem's, in which the soul is literally trapped in the gem and used as a power source.

The Septim dynasty were a line of dragonborn, but there were no dragons to slay as they were thought to be extinct, besided from the mercenary dragon slain in Redguard and that Akaviri Monkey King gone Dragon

So it seems we are in uncharted territory in regards to this form of soul absorption. I wonder if Bethesda will pursue the topic of merging another consciousness into our own in-game at least.

I don't think that Septims were dragonborn, just Tiber Septim (and perhaps Martin, at least the Blades think so).

Actually Tiber Septim got a chance to slay dragon - he slew Ysmir (Dragon of the North), but his soul ended in Mantella.