Europeans have always, or at least as long as this old man can remember, refered to middle eastern people as "asians". As in "Asia Minor". They rarely use the term "middle east".
In the UK, people from the ME are ‘Middle Eastern’ or referred to by the country ie Iraqi, Syrian, Jordanian.
Pakistan is not part of the ME, therefore in the UK are called ‘Pakistani’ or ‘Asian’. Same for Indian or Bangladeshi. The other term we use is ‘south east asia’/’south east asian’.
People from China, Korea, Thailand, Singapore etc are referred to by country or as Oriental or Far Eastern.