This is the way a male chinchilla cleans his privates ... they are very clean animals. If he did not clean himself, he would develop fur rings which could be very bad for him healthwise, and it would be very painful. Nothing to be embarrassed about.
Female chinchillas clean themselves daily, more often when in season, after mating, just before delivery, during and after delivery of the kits.
If chinchillas did not do this, they would not have the reputation they do have of being "almost odorless".
Thinking of cleaning themselves this way may not be appealing, but, they don't use showers/baths and then wash away all their body odor, dirt and waste down the drain to contaminate the water others will be drinking.
They do not touch themselves with their paws to wipe, as humans do ... many of which do not bother to wash their hands and contaminate door knobs, shopping carts and food others eat.
Come to think of it, they are much cleaner than humans.
Jo Ann
P.S. You may want to go out and buy a carry-with-you-size of sanitizing wipes now.