I would definitely get him checked at the vet. It looks like a abscess/infection of sorts. It could be any number of things, but my first thoughts are that of an abscess ... especially if there is any fever at all. It is ALWAYS best to be safe, rather than sorry.
Are you positive it is a male chinchilla ... that almost looks like an abscessed mammary gland.
Are you keeping him isolated from your other chins? If not now, please do so and give him lots of attention ... after you have cared for your other ones.
Wash and sanitize your hands and change clothes after handling your new little one and before handling your others again.
It may or may not be contagious, but it's better to be safe rather than sorry. I purchased a chin from a Michigan breeder that was 'selling off his herd' and ended up with one that could have killed my whole herd, had I put him in with them immediately. I have a very strict rule on isolation with new chins. He was with me two days, when I had to rush him to the vets. It cost me over $800 in vet bills and he died within 5 days after purchase, 3 of which were spent at the vets. What was wrong with him could be passed from one to another by contact or by air. It was a stomach wart that is normally only found in goats and has an incubation period of 9 weeks to 9 months before you know it is there. Even though he had never been in the same room with my herd, I quarantined my whole herd for one year ... just to play it safe. I would not allow any to be adopted during the quarantine time. I was very lucky ... it did not show up ... but it made for a very upsetting and tense year, not to mention an expensive lesson.
The fur chewing could come from irritability, due to what ever is wrong with the swollen area, or just a bit of stress being in a new home.
Is the fur on the back half of his body heavily matted? It looks like it to me.
I would have the vet check him out. Please keep us posted.
Jo Ann