Yes, a DNA swab can link a dog to a particular victim. Here is a link to an article from Scottish wildlife protection group describing a successful prosecution based on finding fox DNA in a dog's mouth via swab. However, the swab must be taken from the dog relatively soon after the bite occurs.
The DNA analysis here is complex, as the DNA of the dog and of anything it has recently eaten must be rules out. Teeth scrapings would be more difficult still, because a scraping would yield a a very high proportion of bacteria, and the amount of the victim's DNA present would be quite small, however it is still technically possible.
Usually in bites cases the DNA matching goes in the other direction; saliva samples are taken from the victim's skin or clothing and are matched to the dog.