Anyone who's emotionally unstable must be diagnosed with something. Right?

Not necessarily. A person may appear emotionally unstable if they are having a healthy, appropriate response to an abnormal situation, or if they were raised in a different culture or a dysfunctional home and did not learn the same social skills as you expect. For example, normal grief may appear "unstable" from the outside. Not everyone whose behavior surprises you has a disease, so best not to judge.