Salt solubility and solvent dissociation
Common ion effect is the key.
A solution of $\ce{NH4OH}$ and $\ce{NH4Cl}$ is basically(no pun intended) a basic buffer solution, which has an almost  constant $\mathrm{pH}$ and thus a fixed concentration of hydroxide ions.
The solubility product of $\ce{Al(OH)3}$ as you say is also constant. So as to maintain the constant nature of both $[\ce{OH-}]$ and $K_{\textrm{sp}}$, there will be a fixed equilibrium concentration of $\ce{Al^3+}$ for a given $\mathrm{pH}$ above which it will start precipitating as $\ce{Al(OH)3}$. By increasing the hydroxide concentration we are forcing the $\ce{Al^3+}$ cation to precipitate earlier than it would have done in a normal, neutral aqueous solution.
This process of forcefully changing the $\mathrm{pH}$ of a solution to alter the solubility of different compounds is very common. In this case it was used to reduce the solubility and precipitate a substance, in other case it might be used to increase the solubility.
For example, during the same qualitative analysis schedule you mentioned, $\ce{HCl}$ is added alongside $\ce{H2S}$ during the detection of group 2 cations(like $\ce{Cu^2+, Hg^2+}$) to prevent precipitation of group 4 cations(like $\ce{Zn^2+, Mn^2+}$) in the form of sulphides. As the acidic conditions increase the solubility of sulphides, added with the fact that group 4 sulphides are not that insoluble prevents them from precipitating with the group 2 cation sulphides.