Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel transport along circles with center at origin depends on the radius of the circle. That is two different circle have different parallel transports.
This question is motivated by this paper.
Yes. Take the Levi-Civita connection of any conformal metric $g = e^{2u}(dx^2+dy^2)$ of positive curvature, say. Then, by (local) Gauss-Bonnet, the holonomy around any smooth closed loop $\gamma$ is of the above form with $a = \cos\theta(\gamma)$ and $b=\sin\theta(\gamma)$, where $$ \theta(\gamma) = \int_{\mathbb{R}^2\setminus\mathrm{Im}\,\gamma} \nu_\gamma\,K\,dA, $$ where $\nu_{\gamma}:\mathbb{R}^2\setminus\mathrm{Im}\,\gamma\to\mathbb{Z}$ gives the winding number of $\gamma$ about the points of $\mathbb{R}^2$ that do not lie on the image of $\gamma$. When $\gamma_r$ is the circle $(r\cos\phi,r\sin\phi)$ for $r>0$, $\theta(\gamma_r)$ is just the total curvature of the metric contained in the interior of the disk. Since $K>0$, this number increases as $r$ increases.
