"small open vessel (smaller than a ship) used to cross waters, propelled by oars, a sail, or (later) an engine," Middle English bot, from Old English bat, from Proto-Germanic *bait- (source also of Old Norse batr, Dutch boot, German Boot), which is possibly from PIE root *bheid- "to split" (Watkins), if the notion is of making a boat by hollowing out a tree trunk or from split planking. Or it may be an extension of the name for some part of a ship.
French bateau "boat" is from Old English or Norse. Spanish batel, Italian battello, Medieval Latin batellus likewise probably are from Germanic languages. Of serving vessels resembling a boat, by 1680s (ship for "serving vessel or utensil shaped like a ship" is attested by 1520s). The image of being in the same boat "subject to similar challenges and difficulties" is by 1580s; to rock the boat "disturb stability" is from 1914.