1560s, from Middle French distinguiss-, stem of distinguer, or directly from Latin distinguere "to separate between, separate by pricking," from dis- "apart" (see dis-) + -stinguere "to prick" (see extinguish, and cf. Latin instinguere "to incite, impel").
The suffix -ish is due to the influence of many verbs in which it is the equivalent of Old French -iss-, ultimately from Latin inchoative suffix -iscere (this is also the case in extinguish, admonish, and astonish). Related: Distinguishing. The earlier form of the verb was distinguen (mid-14c.).