Etymology

prize +‎ fight – the term prize originally referred to rank advancement, not a monetary prize.

Based on Renaissance English Prize Playing (16th century) within the Company of Maisters of the Science of Defence [sic] – public rank advancement fights within the organization, wherein the “prize” was rank advancement. The company later became defunct, while the term was revived in late 17th century for public shows of street fighters, which eventually developed into modern boxing.

In modern times, reinterpreted to mean “a fight for a (monetary) prize”.