I don’t know where the “keep calm” slogan originated, but it certainly applies in the case of fever. Many people consider fever a bona fide emergency, especially in children.
But fever is just a symptom, usually due to a benign (minor) illness that is self-limited (will go away on its own). In less common circumstances, it is due to a serious or even life threatening disease.
“5 truths about fevers in children” explains that, and most of the content applies just as well to adults as children. One difference is that febrile seizures do not occur in adults.
A febrile seizure is one for which there is no other cause – the fever caused the seizure.
So if an older child, adolescent or adult with a fever has a seizure, that seizure must be due to something else; for example, meningitis, an infection of the brain lining, can cause fever and seizures; in that case the seizure is due to the infection, not the fever. In this case, the seizure is the emergency, not the fever.