The most common and most likely answer to this crossword clue is the 12 letter word NAMELESSNESS
N
A
M
E
L
E
S
S
N
E
S
S
Buddy explains!
The straight part
We know that "anonymity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case anonymity = namelessness.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.