Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing.
The city in Somerset, England (Old English Baðun) was so called from its hot springs. Bath salts is attested from 1875 (Dr. Julius Braun, "On the Curative Effects of Baths and Waters"). Bath-house is from 1705; bath-towel is from 1958.
"open wooden vessel made of staves," late 14c., from Middle Low German, Middle Dutch, or Middle Flemish tubbe, of uncertain origin. Related to Old High German zubar "vessel with two handles, wine vessel," German Zuber. Considered to be unrelated to Latin tubus (see tube (n.)); one theory connects it to the root of two based on the number of handles. Also 17c. slang for "pulpit;" hence tub-thumper (1660s) "speaker or preacher who thumps the pulpit for emphasis."
<a href="https://www.etymonline.com/word/bathtub">Etymology of bathtub by etymonline</a>
Harper, D. (n.d.). Etymology of bathtub. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/bathtub