The most common and most likely answer to this crossword clue is the 5 plus 2 plus 5 letter word WHICH IS WHICH
I've seen this crossword clue in The Daily Telegraph
W
H
I
C
H
I
S
W
H
I
C
H
Buddy explains!
The straight part
We know that "A question of double identity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a question of double identity = which is which.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.