There is a fairly recently recognized medical condition called stress-induced cardiomyopathy, associated with chest pain and physical damage to the left ventricle:
Because this weakening can be triggered by emotional stress, such as
the death of a loved one, a break-up, or constant anxiety, it is also
known as broken-heart syndrome.
Although most patients recover within a few weeks, in some extreme cases, this condition is known to be fatal. As this diagnosis is still fairly new, prevalence is not accurately known, but it might be quite common in break ups - see Koulouris et al (2010) for a review.
Psychological distress can be a causal factor in other heart conditions as well (eg, Carey et al 2014; Graff et al, 2016; Buckley et al, 2012) that may be associated with chest pain. However, other than these indirect effects, test subjects can generally distinguish psychological pain from physical pain, so although people may report psychosomatic symptoms such as chest pain associated with romantic rejection, it is not expected to be an indicator.
Nonetheless, across cultures and in many different languages, the same terminology is used to describe both physical and psychological pain, suggesting a similarity in experience.
"Social pain" is the mental stress resulting from rejection, bereavement, heartache, exclusion, etc. There are quite a few studies supporting a link between social pain and physical pain - see Eisenberger (2012) for a fairly recent review. In the typical experimental paradigm, subjects who have recently experienced social pain are reminded of their experience, eliciting the social pain response, while being monitored using physiological and/or neurological measurement tools such as fMRI.
Such studies have demonstrated that social pain experience uses many of the same neurological mechanisms as physical pain. For example, from Kross et al (2011):
Here we demonstrate that when rejection is powerfully elicited—by
having people who recently experienced an unwanted break-up view a
photograph of their ex-partner as they think about being
rejected—areas that support the sensory components of physical pain
(secondary somatosensory cortex; dorsal posterior insula) become
Combined with similar results from lesion studies, pain sensitivity, and changes in dopamine and cortisol, this has led to the hypothesis that humans evolved a mechanism for social pain that recycles the physical pain system (Macdonald & Leary, 2005; Eisenberger & Lieberman, 2004), eg:
It has been suggested that, in mammalian species, the
social-attachment system borrowed the computations of the pain system
to prevent the potentially harmful consequences of social separation.
Overlap between the neural mechanisms for physical and non-physical pain is not complete however, presumably accounting for the differences in reported experience.