The heart is a muscle, therefore it doesn't break it tears (from a puncture, a clogged artery, etc.). If one literally took out their heart and dropped it on the floor, it wouldn't break but simply go "plop".
Stress, however, can cause a heart to malfunction or even stop. Grief and sadness are major stressors.
The "broken heart" most people get is not referring to the muscle, but rather to one's "inner self".
Strangely enough, I have never experienced a "broken heart" (figuratively speaking). I've been sad on occasions, but I never considered that I had a "broken heart".