How to choose the "best" measurment (from a given set) as input for a kalman filter?
Question: Which parameter is suitable to indicate how "good" the
  measurement fits to the Kalman filter?


To estimate a quality of association you can use likelihood function. The likelihood considers not only residual but also uncertainty and represented as scalar value:

$$\mathcal{L} = \frac{1}{\sqrt{2\pi S}}\exp [-\frac{1}{2}\mathbf y^\mathsf T\mathbf S^{-1}\mathbf y]$$


  Problem: For an object tracking scenario with multiple objects and
  multiple tracks, I want to choose the "best" assignment
  object&lt;->track.


The question which you are interested is called data association and as was said there are a lot of solution and this question is too broad. Here are some of methods (you can google it and find more information):
 - Global nearest neighbour (in Euclidean space);
 - Global strongest neighbour;
 - Joint probability data association;
 - Multiple hypothesis tracking (the best choice).