Old English team "descendant, family, race, line; child-bearing, brood; company, band; set of draft animals yoked together," from Proto-Germanic *tau(h)maz (source also of Old Norse taumr, Old Frisian tam "bridle; progeny, line of descent," Dutch toom, Old High German zoum, German Zaum "bridle"), probably literally "that which draws," from PIE *douk-mo-, from root *deuk- "to lead."
Applied in Old English to groups of persons working together for some purpose, especially "group of people acting together to bring suit;" the modern sense of "persons associated in some joint action" is from 1520s. Team spirit is recorded from 1928. Team player attested from 1886, originally in baseball.
mate n.1
mid-14c., "associate, fellow, comrade;" late 14c.,"habitual companion, friend;" from Middle Low German mate, gemate "one eating at the same table, messmate," from Proto-Germanic *ga-matjon, meaning "(one) having food (*matiz) together (*ga-)." For *matiz, see meat. It is built on the same notion as companion (which is thought to be a loan-translation from Germanic). Cognate with German Maat "mate," Dutch maat "partner, colleague, friend."
Meaning "one of a wedded pair" is attested from 1540s. Used as a form of address by sailors, laborers, etc., at least since mid-15c. Meaning "officer on a merchant vessel" is from late 15c.; his duty is to oversee the execution of the orders of the master or commander.