Is there a method to calculate something like general \"similarity score\" of a string? In a way that I am not comparing two strings together but rather I get some number/scores
You might want to look at using a BK-Tree. Here is a discussion and python implementation.
A BK-Tree stores strings in a tree, sorted by Levenshtein distance to the parent nodes. This is normally used to prune the search space when looking for similar strings, but it seems that this tree would form a natural ordering that could be used to create clusters.