Why do we use a Gaussian kernel as a similarity metric? In graph-based clustering, why is it preferred to use the Gaussian kernel rather than the distance between two points as the similarity metric?
Let's be precise. "Distance" has lots of meanings in data science, I think you're talking about Euclidean distance.

The Gaussian kernel is a non-linear function of Euclidean distance.




The kernel function decreases with distance and ranges between zero and one. In euclidean distance, the value increases with distance. Thus, the kernel function is a more useful metrics for weighting observations.
The fact that it's bounded between zero and one is a nice property, whereas the absolute distance (it can be anything) in Euclidean distance can cause instability and difficulty in modelling.
Euclidean distance (without the negative sign) is not a similarity measure, it's a distance function. The gaussian kernel is a similarity measure.
You can think the Gaussian kernel like a normalization function for Euclidean distance.