Why is there no anomaly when particle mechanics is quantized?

Quantum mechanics can also become anomalous. An example is a charged particle moving in a uniform magnetic field. On the classical level, the system is translation invariant in both x- and y-direction. Because the magnetic field is uniform, all (gauge-invariant) measurement will yield the same result at any point of the space, hence the translation symmetry is preserved. But once the system is quantized, the momentum $p_x$ and $p_y$ no longer commute with each other, i.e. $$[p_x,p_y]=\mathrm{i}\hbar B.$$ The non-commutativity is exactly proportional to $\hbar$, implying that this is indeed a quantum effect. In this case, if one chooses to preserve the translation along x, the translation along y must be broken, as $p_x$ and $p_y$ become incompatible observables. This effect is manifested in the wave function under the Landau gauge. Therefore the system becomes anomalous under translation.

Another closely related example is a charged particle moving on a sphere with a magnetic monopole (Dirac monopole) inside the sphere. Let the unit vector $\boldsymbol{n}=(n_1,n_2,n_3)$ be the coordinate that parameterize the position of the particle on the sphere ($\boldsymbol{n}^2=1$). The classical action can be written as a Wess-Zumino-Witten model $$S[\boldsymbol{n}(t)]=\frac{1}{4\pi}\int\mathrm{d}t\int_0^1\mathrm{d}u\;\boldsymbol{n}\cdot\partial_t\boldsymbol{n}\times\partial_u\boldsymbol{n}.$$ The action is invariant under the SO(3) transformation of $\boldsymbol{n}$. But after quantization, the eigenstates are spin-1/2 objects, which are not linear representations of the SO(3) symmetry group. So the system has an SO(3) anomaly.

Anomalies are not particular to quantum field theory, or even to quantum theory. An anomaly is an obstruction to representing some physically relevant group/algebra, often a symmetry group or an algebra of observables, on the state space, and means that your state space will carry not a representation of the symmetry group itself but of an extension. This notion is explained at length in this excellent answer by David Bar Moshe.

Whether the group/algebra that is obstructed is the classical Galilean group that needs the introduction of mass as a "central charge" to become the Bargmann group, the $\mathrm{SO}(3)$ of a particle as in Everett You's answer (which is a special case of a more general link between WZW models and anomalies) that needs the passage to its universal cover $\mathrm{SU}(2)$ on the spin-1/2 which is a central extension by $\mathbb{Z}_2$ or the algebra of fermionic non-Abelian charge densitites that is extended to the Mickelsson-Faddeev algebra (see again the answer by David) by the anomaly term is immaterial - it's all the same principle.

The fundamental character of an anomaly is not a non-invariance of the path integral measure, that's just a particular way to derive it.