Is there a difference what measurement units use in covariance matrix The R matrix in the Kalman filter contains measurement noise. Diagonal elements of the matrix is the power of standard deviation. Is there a difference what measurement unit to use for standard error representation? For instance is there difference between meters and kilometers or radians and milliradians? Should all elements have a similar scale?
Most certainly yes.  The units of the states are reflected in the state covariance, and state error covariance. 

The units of the measurements propagate throughout the filtering and prediction steps.

Should units be appropriately scaled? Numerical accuracy typically benefits from appropriate scaling