They are somewhat special. Think of it as an Untargetable Zyra plant with all the negative effects of auto attacks.
The soldier itself is definitely some form of untargetable and invulnerable minion (As I said like an invulnerable Zyra plant or heimer turret).
Now the attacks of the soldier are a mixture of auto attacks and spells. The player himself has to command the attacks directly by clicking on a target in range. Note that your Azir will stop moving if he is out of range but the soldier is able to attack the target while he will just walk up to the enemy target to attack if there is no soldier near. The fact that this is triggered by a right click (or attack move) against a target implies that they count as an auto attack in terms of minion aggro.
But then again the attack originates from the sand soldier and not from Azir himself. This allows it to be a spell that applies on-spell effects (Hello Zyra Plants).
TL;DR
The Soldiers are untargetable minions, The attacks count as Auto Attack for Azir and as spells for the soldiers.