Asymptotically conical shrinking Ricci solitons are an important class of potential singularity models for the Ricci flow in dimension four and higher, and give rise to solutions which, near infinity, flow smoothly into a cone at the singular time. We will present some uniqueness and nonexistence results which can be inferred from this latter fact using a unique continuation result valid at the singular time, and discuss their applications to the classification problem for asymptotically conical shrinkers.