Although the latter term is less publicized in the mass media, a WHO page basically calls it official too, although it was the ICTV that came up with it:
Viruses are named based on their genetic structure to facilitate the development of diagnostic tests, vaccines and medicines. Virologists and the wider scientific community do this work, so viruses are named by the International Committee on Taxonomy of Viruses (ICTV).
Diseases are named to enable discussion on disease prevention, spread, transmissibility, severity and treatment. Human disease preparedness and response is WHO’s role, so diseases are officially named by WHO in the International Classification of Diseases (ICD).
ICTV announced “severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)” as the name of the new virus on 11 February 2020. This name was chosen because the virus is genetically related to the coronavirus responsible for the SARS outbreak of 2003. While related, the two viruses are different.
WHO announced “COVID-19” as the name of this new disease on 11 February 2020, following guidelines previously developed with the World Organisation for Animal Health (OIE) and the Food and Agriculture Organization of the United Nations (FAO).
WHO and ICTV were in communication about the naming of both the virus and the disease.
So yeah, the two names originate from different organizations (as far as I can tell the ICTV is not subordinated to the WHO in any way), but they apparently held some talks on the matter. Nonetheless, the WHO does seem a little displeased with including "SARS" in the virus name...
From a risk communications perspective, using the name SARS can have unintended consequences in terms of creating unnecessary fear for some populations, especially in Asia which was worst affected by the SARS outbreak in 2003.
For that reason and others, WHO has begun referring to the virus as “the virus responsible for COVID-19” or “the COVID-19 virus” when communicating with the public. Neither of these designations are intended as replacements for the official name of the virus as agreed by the ICTV.
Likewise, some virologists were displeased with the WHO name. In the later published SARS-CoV-2 (naming) paper of ICTV's CSG (Coronaviridae Study Group) they do "mend fences" however, saying that:
By uncoupling the naming conventions used for coronaviruses and the diseases that some of them cause in humans and animals, we wish to support the WHO in its efforts to establish disease names in the most appropriate way. [...] The further advancement of naming conventions is also important because the ongoing discovery of new human and animal viruses by next-generation sequencing technologies can be expected to produce an increasing number of viruses that do not (easily) fit the virus–disease model that was widely used in the pre-genomic era [...]. Having now established different names for the causative virus (SARS-CoV-2) and the disease (COVID-19), the CSG hopes that this will raise awareness in both the general public and public health authorities regarding the difference between these two entities. The CSG promotes this clear distinction because it will help improve the outbreak management and also reduces the risk of confusing virus and disease, as has been the case over many years with SARS-CoV (the virus) and SARS (the disease).