Is it correct to assume that PageRank is a sorting algorithm or does it fall in any other category?
The answer depends on what you mean by a sorting algorithm. The usual meaning is:
A sorting algorithm is an algorithm that gets a list $L$ of comparable objects, and sorts them in (say) nondecreasing order. The algorithm could only work for a certain class of objects.
If you use this definition, then PageRank isn't a sorting algorithm. But it's up to you.