I have provided my answer already in another question.
https://stackoverflow.com/a/30120166/2282794
I have worked on really large scale system with similar name matching requirements that you have talked about.
Name matching is not very straightforward and the order of first and last names might be different.
Simple fuzzy name matching algorithms fail miserably in such scenarios.
If we just want to talk about the Approximate String matching algorithms, then there are many. Few of them are: Jaro-Winkler, Edit distance(Levenshtein), Jaccard similarity, Soundex/Phonetics based algorithms etc. A simple googling would give us all the details.
You can implement all of them in C#
Irony is, they work while you try to match two given input strings. Alright theoretically and to demonstrate the way fuzzy or approximate string matching works.
However, grossly understated point is, how do we use the same in production settings. Not everybody that I know of who were scouting for an approximate string matching algorithm knew how they could solve the same in the production environment.
I might have just talked about Lucene which is specific to Java but there is Lucene for .Net also.
https://lucenenet.apache.org/