What distance metric can I use for comparing image features like elongation and solidity of each image?
If I understood you correctly, each contour is described by a 2-element feature vector $f = [e, s]$, where $e$ is elongation and $s$ is solidity.

In that case, you might want to try the Mahalanobis distance, which is defined as follows:
$$d(f_1, f_2) = \sqrt{(f_2 - f_1)C^{-1}(f_2 - f_1)}$$
where $f_1$ and $f_2$ are the feature vectors that you are comparing, and $C$ is the covariance matrix of your data set.