How many times has SARS-CoV-2 mutated?
This question makes a number of incorrect assumptions and I don't have time to correct them. The short answer is that the virus has mutated probably hundreds of times since it entered humans in late 2019. 

The lower figure on the NextStrain.org ncov page, "Diversity", shows the known mutations that have been identified so far. As I look at it now, there are maybe 100-200 shown there, but that will change daily and I can't be bothered to count them.

Is it possible the virus has mutated twice in a period of time of months? Of course it is. That's what we expect from coronaviruses. It would be shocking if it did not. 

The default assumption (based on a vast amount of experience with coronaviruses and many other viruses) is that these mutations are neutral and do not affect the virus in terms of fitness, virulence, or transmissibility in any way; they are occasionally useful in tracking sources, but unless you have spent a lot of time looking at virus phylogenetic trees for several years, your interpretations of these mutations are almost certainly wrong.

An example of people making claims based on unfamiliarity with virus evolution is the "two different types of the virus" claim. That claim is addressed by MacLean and colleagues in their Response to “On the origin and continuing evolution of SARS-CoV-2”. Summary:


  Two of the key claims made by this paper appear to have been reached by misunderstanding and over-interpretation of the SARS-CoV-2 data, with an additional analysis suffering from methodological limitations