What is the reason why anyons escape spin-statistic theorem?

You are right. In a space-time with one time dimension and $D$ spatial dimensions, finding possible different statistics is equilalent to look at the fundamental group (first homotopy group) of $SO(D)$

For $D=1$, the fundamental group is trivial.

For $D=2$, the fundamental group is $\mathbb{Z}$.

For $D>=3$, the fundamental group is $\mathbb{Z}_{2}$

So, it explains, why, in 3 spatial dimensions, there are only 2 kinds of statistics (fermions and bosons), while the situation is different with 2 spatial dimensions.

For quantum point of view, you will have to find unitary representations of this fundamental group.