Why ammonium hydroxide precipitates metal salts?

Zinc has amphoteric properties like aluminum. It can dissolve in bases as well as acids. In bases it forms zincate anions. The "why" part of science often has no answer because the reason is that nobody fully understands such a complex phenomenon. It is relatively easier to talk about the observational part.

An aqueous solution of ammonium is very basic, which means that there are free hydroxide ions there. Writing NH3.H2O is deceptive because many general chemistry texts emphasize that NH4OH does not exist and they advocate NH3.H2O. One way to rationalize the observation of the Ksp values for zinc hydroxide i.e. if we have free zinc ions and free hydroxide ions, a solid Zn(OH)2 is favored at that particular pH.

It's simply because of the concentration. In aqueous solution, the formation of a precipitate is favored over the formation of a complex ion, and it can be shown using solubility equilibria and $K_mathrm{f}$ (but it won't be pretty).

This is just like the QA test for copper(II). After a small amount of aqueous ammonia is added, $ce{Cu(OH)2}$ is observed. Add more, and you get the deep blue $ce{[Cu(NH3)4]^2+}$ complex.