Think of a body running too hot: skin burning, heart quick, unable to lie still. Feverish holds that one picture. Sometimes the heat is real illness. Sometimes it is the heat of wanting, worrying or hurrying, and the body behaves the same way.
Wiktionary records the word as fever + -ish, that is, 'having the look of a fever'. The illness sense is the plain reading; the excitement sense reuses the same look.
Wiktionary 記的組成是 fever（發燒）加 -ish（帶有…樣子的）。字面就是「帶著發燒的樣子」。真的發燒是最直接的一種樣子。心裡的燒是把同一個樣子借去用。
Wiktionary entry for feverish, packet evidence kaikki:3f6f69a190036ae4:1101918783:17723 — From fever + -ish.
Two branches
身體真的在發燒
這是核心畫面最直接的一種：熱是真的，來自生病。
adjective · B2
Having a body temperature that is higher than normal because you are ill, so that you feel hot, weak and uncomfortable. It is also used when you only feel that way, before anyone has checked your temperature.
發燒的
像發燒一樣的激動與忙亂
熱從身體移到心裡：沒有人生病，但那種靜不下來的樣子一模一樣。
adjective · C1
Full of excitement, worry, hurry or wanting that is so strong it makes people move, work or feel much faster than usual, as if they could not stop.
狂熱的；焦急忙亂的