This is a pretty common place for chins to chew on themselves. Sometimes it can indicate that they are in pain, but usually it's just something they do. I've seen chins chew themselves or have others chew them on the tail right near the base or down the top of the tail, or even more of the fur. I've had chins that seem to trim the fur on their own tails or the tails of others, as strange as that sounds.
I have an older female, she's 15 now. She decided one day after never fur chewing or anything that she wanted to chew all the fur off of her tail, just off her tail. She's not a very anxious chin, and she's always chewing on toys and playing with her friend...it makes no sense except that she started chewing one day and really enjoyed it.
Unfortunately, I think that with your chinchilla it's a behavioral issue. I don't think there's much you can do. The only thing that anyone can do to help this type of problem is to provide more toys and more things to keep the chin busy. I know that it can be caused by stress, or appear to be caused by nothing at all. I had a young male that did it for a few months last year and then he stopped chewing at all.
As far as I know this type of chewing doesn't at all appear to be caused by nutritional deficiency. When chins chew on their coats, it can definitely be from improper diet. For some reason just chewing on the tail is usually different. The chin can be very healthy, be completely happy and still chew off that little strip of fur off the top of the tail.
I wonder if the vet will have more insight for you. How has your chin been behaving recently? Has he been eating? Does everything but this seem to be normal?
