shower bath vs bathing

shower bath and bathing both are nouns.

Word	Noun	Adjective	Verb	Adverb
shower bath	Yes	No	No	No
bathing	Yes	No	No	No

As nouns, bathing is a hypernym of shower bath; that is, bathing is a word with a broader meaning than shower bath:

shower bath: washing yourself by standing upright under water sprayed from a nozzle
bathing: the act of washing yourself (or another person)

Other hypernyms of shower bath include washup.

shower bath (noun)	bathing (noun)
washing yourself by standing upright under water sprayed from a nozzle	immersing the body in water or sunshine
booth for washing yourself, usually in a bathroom	the act of washing yourself (or another person)