Without the B note, in current naming of chords, the chord would usually be called A5. B is the 9th 0r 2nd of the A scale, so A5add2 or A5add9 would be unambiguous.

I don't like A5 as a name, but it's commonly used in rock and pop.

Add9 as noted above is used instead of just '9' because the normal usage of '9' includes the dominant 7.

As to whether it should be called a chord because it doesn't conform with older rules of music, I think it's a losing battle. It is a chord in current usage.

Stu