I think the most relevant (yet also quite weak) commonality between osteopathic medicine and EMS is the focus on primary care medicine. I noticed that my EMT experience helped me very much in my primary care skills course. I assume that if I continue on to a primary care or other initial point-of-contact speciality, much like the rest of the decreasing majority of DO graduates, that my background in EMS may be beneficial.
Again, a weak commonality.