The most common and most likely answer to this crossword clue is the 6 plus 6 letter word SERIAL NUMBER
I've seen this crossword clue in Independent
S
E
R
I
A
L
N
U
M
B
E
R
More clues leading to the result SERIAL NUMBER
Buddy explains!
The straight part
We know that "Identifying feature of anaesthetist?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case identifying feature of anaesthetist? = serial number.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.