Three Metrics, Three Different Neighbours

All three metrics pick $A$: it agrees with $\mathbf{q}$ exactly in two of the three coordinates, so no other point can be closer however you measure.

Manhattan picks $A$, Euclidean picks $B$, and Chebyshev picks $C$ — all three metrics name a different neighbour.

All three metrics pick $B$, which is the point closest to the origin and therefore the point closest to $\mathbf{q}$.

Manhattan picks $A$, Euclidean picks $A$, and Chebyshev picks $C$ — only Chebyshev disagrees.