Yes, it's the present perfect continuous.
The use of the present perfect continuous focuses on a more extended period of time than the present perfect simple, and on the repetition of actions during that period.
If you say 'Why haven't you answered my letters?', you are concentrating on one action - on the one thing which the other person has so far failed to do. The implication is that you only expected one reply, but you haven't received this.
'Why haven't you been answering my letters?' focuses on a whole period of time, and the events of this period. During this period you have been sending letters repeatedly and the other person has repeatedly been ignoring them. The implication is that you expected the person to reply to each of your letters.