I don't think what causes kennel cough has anything to do with S. equi. I had always thought that cross species transfer couldn't occur with S. Equi, but out of curiosity I did some searching and, to my surprise, found that it can…and not all cases can be traced back to an animal as the cause of infection.
References: Group C/G strep - The Lee Spark Necrotising Fasciitis (NF) Foundation
"Despite the link to animals, few people with invasive group C or G streptococcal illness have any history of contact with farm animals or horses. The vast majority of group C and G disease is picked up in the community - they are not normally considered to be ‘hospital infections’, even in patients following surgery." "Horse strep" in people : Worms and Germs Blog
"Streptococcus is a group of bacteria that includes many different species. There are two main species in horses Streptococcus equi subsp. equi*(aka S. equi, the cause of strangles) and Streptococcus equi subsp. zooepidemicus (aka S. zooepidemicus, a cause of various types of infections). As you can guess by the 'equi' name, their natural host is the horse. Strep infections are very common in people, but rarely involve these two species. Nonetheless, infections with either Streptococcus equi or S. zooepidemicus can be found in people, but S. zooepidemicus is most common. Usually, these infections develop in people who are already sick for another reason, have compromised immune systems, or in young children. Interestingly, not everyone that is infected reports direct or even indirect contact with horses."
And, from the book Equine Infectious Diseases by Deba C.Sellon, Maureen T. Long, pg 256 under subhead "Public Health Concerns" last sentence reads:
"S. Equi is highly host adapted, however, and infections of humans have rarely been confirmed." LA County Department of Public Health-Veterinary Public Health - Equine Strangles
Transmission to Humans. In rare cases, humans have contracted infections from the bacteria that cause Strangles.* To prevent human infection, people caring for horses with Strangles should avoid getting any nasal or abscess discharge from the horse on their eyes, nose, or mouth.* The should also wear disposable gloves while working with the horse, avoid touching their face, and should wash their hands thoroughly when finished.*"
"Human infections with Strangles, although not common, do occur."
Links to scientific articles on human cases
2003 - Streptococcus equi subspecies equi (Lancefield group C) meningitis in a child
2006 - Post streptococcal acute glomerulonephritis secondary to sporadic Streptococcus equi infection
2004 - Primary purulent pericarditis due to group C Streptococcus (These links are located at the bottom of the publichealth site above)