Synesthesia is a general term for conditions in which a person does not experience sensory qualia in the usual way -- generally, attaching one set of qualia to a set of experiences usually associated with another, e.g., experiencing tastes when hearing words. Mirror-touch synesthesia is a type of synesthesia in which a person feels tactile sensations matching that of a person they are observing. And pain synesthesia is a subset of mirror-touch synesthesia in which a person experiences pain when they see another person being hurt.
Things become fuzzy here, as pain empathy is a core part of human empathy; we notice and care when others are being hurt. This sometimes expresses itself, internally, as a 'wince' in a corresponding body part to the part you see being hurt. There is a somewhat unclear continuum between localized pain empathy and pain synesthesia.
Various people report a paired tingle, 'electric pulse', weak feeling, muscle tension, or the like when seeing another person experience pain; even more people report a somatic response that is not localized to the point that the other person is feeling pain, most often a 'stomach drop', tension or pulse in the abdomen, or other abdominal response, including nausea. However, the presentation of pain empathy is as varied as are human brains, and this is an overview of what is most common, and cannot be an exhaustive list.
Various studies have looked at different levels of pain empathy and pain synesthesia, without a united model of how they would define these terms; they tend to have very small sample sizes, and many cases are simple case studies. For example, it has been noted that people who suffer from phantom limb syndrome are particularly prone to feeling emphatic pain in their phantom limbs; this has naturally been a very hard phenomenon to study rigorously.
Very roughly speaking, about 15-30% of the population have some form of pain empathy, and some smaller percent experience pain synesthesia. However, this is a situation where defining and communicating the terminology is difficult enough that we may never have accurate numbers; human brains are just too diverse.