The online search for 'anger anguish etymology' took me further than I'd expected. Etymonline.com states that the two words (and a few others) share a common ancestor, the Proto-Indo-European (PIE) root 'angh-' which means 'tight, painfully constricted, painful'.
This common ancestor also forms all or part of: angina, angst, anxious, hangnail and quinsy. All these words are talking about a constriction, a tightness.
Here is the entry for 'anguish' (n.)
C. 1200, "acute bodily or mental suffering," from Old French anguisse, angoisse "choking sensation, distress, anxiety, rage" (12C), from Latin angustia "tightness, straitness, narrowness," figuratively "distress, difficulty," from ang(u)ere "to throttle, torment" (from PIE root 'angh-' "tight, painfully constricted, painful").