Why is 2-(3-aminomethyl-2-hydroxypropyl) cyclohexan-1-one named so and not 2-(2-hydroxy-4-aminobutyl) cyclohexan-1-one?
As drawn the compound does not contain a butyl side chain with four carbon atoms; the side chain is only a propyl chain with three carbon atoms.

The propyl side chain has two substituents: a hydroxyl ($\ce{-OH}$) group and an methylamino ($\ce{-NH-CH3}$) group. Since these groups are not the principal characteristic group in this case, they have to be expressed as prefixes. Note that the correct prefix for the secondary amine is “methylamino” and not “aminomethyl” as proposed in the question.

Therefore, the correct name is 2-[2-hydroxy-3-(methylamino)propyl]cyclohexan-1-one.