Aurelion Sol, primarily seen as a caster, can actually be potentially played in a multitude of roles or lanes.

Mid
Aurelion Sol is most likely to be seen mid lane, due to his mage/caster-like kit; and in which case is as valid as any other caster (poke and bully out of lane, be aware of assassins in close range, etc.)

Bot - APC
While Aurelion Sol is popular in literally every lane except bottom, I digress.
Firstly, lets take a look at what an ADC accomplishes:
- Sustained damage from the backline (Done by Sol's passive alone)
- The ability to kite and chase (Sol having a stun, ult, move sphere length)

Let's also look at the advantages Sol has over other ADC's
- Can build AP-offtank items (Rod of Ages, Liandry's, Rylai's) to be tankier than other ADCs
- Has a large AOE stun and knockback, being useful in fights other than damage
- Can constantly move and reposition during the fight, without having to stop and AA