In the field of Data Mining, is there a specific sub-discipline called 'Similarity'?
Yes. There is a specific subfield in data mining and machine learning called metric learning, which aims to learn a better distance metric among data instances.
Do you know any of the following concepts?
Euclidean distance
Mahalanobis distance
Pearson correlation
Cosine similarity and here
Kernel functions
After you know these, you will know what is 'similarity'.
I would like the community opinion on how closely related Data Mining and Artificial Intelligence are.
It is very hard to distinguish what is data mining, what is AI. Don't discuss this question when you are new in the field. When you have learned 10 algorithms in data mining and read some AI books, you will know the difference and the relation.