Is Neural Style Transfer considered supervised or unsupervised learning? As far as I understand it, the Neural Style Transfer uses a content image and a style image, and generate a new image based on the two images.

It tries to find a set of pixel values such that the cost function J(C, S) is minimized. It does not have any labels associated in advance, but it has an output (generated image) that should be the target of the learning.

However, I'm not sure if this is considered supervised or unsupervised learning. Which does it belong to?
Neural style transfer is not really machine learning, but an interesting side effect/output of machine learning on image tasks. When performing neural style transfer using a pre-trained model, then a significant amount of supervised machine learning has already occurred to enable it.

The style transfer algorithm is still an example of gradient-based cost function optimisation, which it shares with many supervised and unsupervised learning algorithms.

The output of style transfer is partially a probe into what a network has learned about different levels of structure in the problem domain it has been trained on. However, its main use has been to generate images with altered/mixed aesthetics for art and entertainment.