There are a two levels for the answer:
First, perturbative string theory actually has two different scales: the string scale in which the extended nature of the string comes into play (even classically strings are different than particles), and the Planck scale in which the quantum mechanics of the string (including quantum gravity) becomes important. The string energy scale is lower than the Planck energy scale, their ratio is the string coupling constant (which is proportional to $\hbar$), so it is meaningful to talk about stringy physics in the sub-Planckian regime. Your mental picture then depends on the size of the compact geometry.
When the CY manifolds are larger the Planck length you can use perturbative string theory, in which case the notion of a manifold still kind of makes sense. When reaching the Planck scale the string theory is no longer perturbative and large fluctuations in the geometry are part of the story. Since we know quite a bit about non-perturbative string theory (through dualities and non-renormalization theorems) these expected phenomena can be observed and analyzed precisely. For example, topology change of the CY manifold is a smooth process that can be discussed very precisely in this framework.
(Incidentally, in this context we see that in the transition region between two well-defined geometries there is no geometrical description at all - contrary to some people's intuition about quantum geometry and spacetime foam and all the rest of that. The geometrical description fails and there is some well-defined procedure to do any well-defined calculation without any reference to any type of geometry whatsoever).
The second level of answer is that even when discussing perturbative strings on sub-Planckian CY manifolds, you have to keep in mind that this is a shorthand for something more precise and technical - (2,2) SCFT which has a limit in which it becomes a sigma model with a CY manifold as a target space. In English this means that even classical strings in some target space probe the geometry in different ways than point particles, and to describe their physics you need more complicated machinery than just differential geometry. As you suspected the fact that the string is extended means that it is more forgiving to non-smooth features of the manifold, even classically. This is encoded in the fact that the SCFT is better behaved than differential geometry on that space, and stays well defined even in the presence of some types of singularities (orbifolds of space are a famous example).
So, in summary "string theory on CY" really means string theory whose classical point particle limit (valid when the manifold is very large and smooth) reduces to CY manifold. Good to remember in general that there is some poetic license taken sometimes when physicists describe their work.