The compression method to assess similarity, in the sense of having a small normalized compression distance (NCD), was developed based on algorithmic information theory to quantify the similarity in files ranging from words and languages to genomes and music pieces. It has been validated on objects from different domains always using essentially the same software. We analyze the whole-genome phylogeny and taxonomy of the SARS-CoV-2 virus, which is responsible for causing the COVID-19 disease, using the alignment-free compression method to assess similarity. We compare the SARS-CoV-2 virus with a database of over 6500 viruses. The results suggest that the SARS-CoV-2 virus is closest in that database to the RaTG13 virus and rather close to the bat SARS-like coronaviruses bat-SL-CoVZXC21 and bat-SL-CoVZC45. Over 6500 viruses are identified (given by their registration code) with larger NCDs. The NCDs are compared with the NCDs between the mtDNA of familiar species. We address the question of whether pangolins are involved in the SARS-CoV-2 virus. The compression method is simpler and possibly faster than any other whole-genome method, which makes it the ideal tool to explore phylogeny. Here, we use it for the complex case of determining this similarity between the COVID-19 virus, SARS-CoV-2 and many other viruses. The resulting phylogeny and taxonomy closely resemble earlier results from by alignment-based methods and a machine-learning method, providing the most compelling evidence to date for the compression method, showing that one can achieve equivalent results both simply and quickly.