Structuring Kalman filter for tracking problem where only position is known
Just a slight nit $\sigma^2$ is a variance $\sigma$ is a standard deviation.

Let me answer your last question first.  In a KF, one has measurements and states, and they are not usually the same.  In a Savasky Golay filter, one is estimating the derivatives of the measurement, not the state.  In your case the measurements are trivially related to the states.  

I'm guessing but I think you want something like
$$ 
\left| \begin{array}{c} x \\ y \\v_x \\ v_y \end{array}\right| =
\left| \begin{array}{cc}
1 &amp; 0 &amp; \Delta_t &amp; 0 \\
0 &amp; 1 &amp; 0 &amp;\Delta_t \\
0 &amp;  0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 0 \end{array} \right| +
\left| \begin{array}{c} 0 \\ 0 \\ w_1 \\ w_2 \end{array} \right|
$$
for the state model  $\Delta_t$ is your time step and $w_1, w_2$ iid Normal noise.

If this model is your problem,  yes the KF is what is wanted.

What you will need to figure out, is what is process noise and what is measurement noise.

A KF needs measurement noise to work, process noise is optional, but you would need a different state model if you want to just use measurement noise.