Catboost multiclassification evaluation metric: Kappa & WKappa
I posted the same question on the Catboost github (issues) page and got an answer.
The link can be found here: https://github.com/catboost/catboost/issues/1447
Answer:
Class weights and weights in WKappa are different.
Calculation of WKappa metric consists of two steps:

Confusion matrix calculation - here object and class weights are used.
WKappa calculation using confusion matrix - here values from confusion matrix multiplied by squared difference between classes. (i-j)^2.
In other words, weight of class 0 has the same effect when object of class 0 was classified as class 1 and when classified as class 5. But WKappa weight for misclassification 0-&gt;5 will be 25 times more than for misclassification 0-&gt;1.
So you can use both class weights and WKappa if your classes are ordinal.