Why the Silhouette Score and optimal number of Cluster changes when using 2D and 3D data?
Yes it can happen. In fact it is quite normal since there are different clusters in 2D and different in 3D, since more or less information is added to data (by having more dimensions). This is a by-product of the curse of dimensionality.
Adding as more relevant information as possible would make clusters more close to underlying groups. So 3D would be better than 2D. This is a general observation. There are of course cases where projecting data in a low-dimensionalal manifold is indeed better since it can eliminate noise and/or capture specific attributes better than clustering on all (possibly irrelevant) dimensions (another by-product of the curse of dimensionality).
If the relevant information in your data has low dimensionality but this information is correlated along many dimensions in the original data then a feature extraction method is needed in order to capture the low-dimensional relevant information from original data (eg PCA, ICA ,..).
For some references along this direction see for example:

How to cluster in High Dimensions
An investigation of K-means clustering to high and multi-dimensional biological data
How do I know my k-means clustering algorithm is suffering from the curse of dimensionality?