Does the Coleman-Weinberg mechanism belong to the dynamical symmetry breaking or the anomaly?

First, dynamical symmetry breaking (which I take to be either synonymous with or a subset of spontaneous symmetry breaking) and anomalies are two completely different things. An anomaly is when a symmetry group acquires a central extension, due to some obstruction in the process of representing it in our theory. Such obstructions can exist purely classical, or they can arise in the course of quantization, but they are crucially features of the whole theory. For more information on anomalies, see this excellent answer by DavidBarMoshe. In contrast, in spontaneous symmetry breaking, the theory retains the symmetry, just its vacuum state does not, which leads to the symmetry being non-linearly realized on the natural perturbative degrees of freedom (being "broken").

Just $\phi$ acquiring a VEV would not mean an anomaly, that would just be ordinary spontaneous symmetry breaking. However, the appearance of the $\phi^2$ term in the effective potential also means that we have an anomaly, i.e. the quantum effective action is not invariant under the classical symmetry - this is a clear case of a quantum anomaly. That is, in this case, the Coleman-Weinberg mechanism leads to both spontaneous symmetry breaking and a quantum anomaly, but it is perfectly conceivable to have one without the other - they are completely distinct things. It might be debatable whether we want to speak of spontaneously "breaking" a symmetry that became anomalous to begin with, though.

Strictly speaking, the phase dynamical symmetry breaking, although essentially a kind of spontaneous symmetry breaking proces, means something slightly different. While spontaneous symmetry breaking usually refers to cases where an elementary scalar field (such as $\phi$) acquires a VEV, dynamical symmetry breaking refers to cases where the scalar field that acquires a VEV is a composite field. In this sense the term dynamical refers to a force that is strong enough to bind fields so strongly that the composite acquires a VEV. Chiral symmetry breaking is an example of dynamical symmetry breaking. Technicolor theories are attempts to describe the electroweak symmetry breaking in terms of a dynamical symmetry breaking process.