Generally, I think it is important to know the definitions of technical terms and to use them properly. However, perhaps in this case it is not too important if a rotating "object" which is a pure mathematical quantity having a constant magnitude and a changing phase (proportional to time) is called "vector" or "phasor".
As already mentioned, the inventor of the name "phasor" did consider this rotating object as a kind of vector. Other authors restrict the term "vector" to purely geometric objects.
For my opinion, more important is to know what this quantity means and how it is used in system theory.