I do not recall the annual flu shot ever being called a flu “vaccine” until we were talking about COVID-19 vaccines.Miriam Webster’s online dictionary added an additional definition to include the mRNA shot for COVID-19
Here is the information from the CDC website posted in 2019, before we knew about COVID:
“For the 2019-2020 flu season, ACIP recommends annual flu vaccination for everyone 6 months and older with any licensed, age-appropriate flu vaccine (IIV, RIV4, or LAIV4) with no preference expressed for any one vaccine over another.”
It is true though that lots of people call it the “flu shot” — probably because two syllables is easier to say than three. A lot of people call it the “COVID shot”, or the “COVID jab”.