That's because the whole of the Middle East is in fact a part of the Asian continent. Look at a map of the continents. This comes up every other day here, it seems. I'm surprized some long time posters are still puzzled by this (the poster you responded to, not you).
So the Brits use a designation based on continents, not 'regions'. Big deal.
The British and Irish see a clear difference between the ME and Asia, be it SE Asia or the Far East. Therefore we do not include Pakistan for example as part of the Middle East, as Americans do. To us, it is South East Asia.
Personally, I prefer the term "middle eastern" only because then you can tell for sure that they are talking about pakistanis vs. chinese for instance. But that is just me.....
But as you said, more people need to recognize that other people have varying uses for terms...