Part 1: Dragons.

I think that souls are made of two separate components. One part is the souls energy, the other it's personality and memory. For dragons these parts separate easily after death. In addition, their energy is magnetic. Live dragon energy attracts dormant dragon energy. If a dragon or dragonborn doesn't or can't resist then the energy is absorbed. The thu'um can be learned this way as the deeper meaning and "shape" of words of power are permanently imprinted on the souls essence. Once a dragon's soul energy has been split from the persona it can never be restored, preventing ressurection. What happens to the persona can only be speculated on. It is likely that most dragons refrain from absorbing their fallen brethrens essence, but that the dragonborn doesn't know how to do this and absorbs it automatically.

Part 2: Mortals.

With mortals, the soul typically remains intact after death, either moving on to the after life or remaining in a spectral form. The exception is when the soul is trapped inside a soul gem. Using the soul gem the splits the soul, sending the persona to the Soul Cairn and using it's energy to power magical items. 

Part 3: Dragonborn.

Dragonborn are unique. Their souls are made from dragon energy with a mortal persona. This gives them the ability to shout from instinct and absorb a fallen dragon's soul energy. It seems likely that upon death their soul would also split, allowing their mortal persona to ascend to the after life while the dragon energy remains behind to be absorbed by another. This also explains why dragonborn don't feel any distress about having a mortal form as the part of their soul tied to their form (the persona) is mortal and not dragon.