Degree comes to Early Modern English in the 13th century from the French word degré (first appearing in French a century earlier) which in turn comes from the Latin word gradus meaning position, gait, pace or step. It has been used to indicate even divisions (other than steps) since antiquity-use in temperature comes from 1727. Early modern English used the etymologically related grade to indicate temperature divisions, and Anders Celsius called his temperature scale the Centigrade scale, to indicate 100 regular divisions or degrees.
Outside of science, the word degree has a variety of uses: in mathematics it indicates the regular divisions of a circle into 360 equal parts; in criminology it has been used since the 16th century to indicate the heinousness of a crime; in education it has been used for hundreds of years to indicate posession of teaching or other qualifications.
Image of a vintage mercury thermometer courtesy Pieter Kuiper-image released into the public domain.