Think of the ending -er as a label you can stick on anything: the one that does this. Stick it on fight and you get the one that fights. Everything else follows from what you stick it to. Stick it on a person in a ring and you have an athlete. Stick it on a person in a war and you have a soldier. Stick it on a small quick plane and you have the machine that goes after other planes. Stick it on a game and you have the genre. Stick it on someone facing a hard year, and the fighting is no longer literal at all.
The evidence packet records fighter as Middle English fightere, feghtere, from Old English feohtere, and analyses it as fight plus the agent ending -er. The modern uses are all that same formula applied to different things.
來源資料記載 fighter 來自中古英語 fightere、feghtere，再往前是古英語 feohtere，構造上就是 fight 加上表示「做這件事的人或東西」的 -er。今天的各種用法，都是同一個公式套在不同對象上的結果。
Wiktionary entry for fighter, etymology_text supplied in this evidence packet (source_ref kaikki:3f6f69a190036ae4:282648058:58474)
Three branches
動手去打的人
最字面的一組：-er 貼在人身上，打的動作也是真的。差別只在場景是運動場還是戰場。
noun · B1
A person who fights other people, especially as a sport — a boxer, a judo or taekwondo competitor, a mixed martial arts athlete. It is also the plain word for anyone taking part in a fight.
格鬥選手；打鬥的人
會打的東西
同一張 -er 標籤改貼在物件上。動作一樣是打，只是執行的不是人。
noun · B2
A small, fast military plane built mainly to attack other planes in the air. Often used before another noun: a fighter pilot, a fighter jet.
戰鬥機
比喻：撐住不放棄的人
標籤還在人身上，但「打」變成比喻。對手不是另一個人，而是病痛、時間或一連串的拒絕。
noun · B2
Someone who keeps trying and does not give up, even when things are very hard. Often said warmly about a person going through illness or a long, difficult stretch of life.
不放棄的人；鬥士