While "quantizing" a classical theory (such as that of a harmonic oscillator), what we do is that we try to find (in particular, guess) a consistent quantum theory whose (one) classical limit would be the classical theory with which we started out.
I don't think there is a distinction between "a quantum harmonic oscillator" and "a classical harmonic oscillator at a quantum scale". In particular, because, the only thing one can possibly mean with "a classical harmonic oscillator at a quantum scale" would be a quantum theory which looks like the classical harmonic oscillator in (one of) its classical limit(s)--and that is precisely what we mean by "a quantum harmonic oscillator" as well because we invented the word "harmonic oscillator" in a classical context and when we say "a quantum harmonic oscillator", we precisely mean a quantum theory which looks like a classical harmonic oscillator in (one of) its classical limit(s).
One thing to keep in mind would be that a quantum theory is more basic and thus, it might often be more instructive to think about how a classical theory emerges from a quantum theory than the other way around. Because while we might use a classical theory to second guess a quantum theory out of which the known classical theory emerges, nature certainly works the other way around. Moreover, it might also be useful to notice that there can be quantum theories which admit no, one, or more than one classical limits.