bathing vs shower

bathing and shower both are nouns.

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

As nouns, shower is a hyponym of bathing; that is, shower is a word with a more specific, narrower meaning than bathing:

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

Other hyponyms of bathing include shower bath, bath, sponge bath, steam bath, Turkish bath, vapor bath, vapour bath.

bathing (noun)	shower (noun)
immersing the body in water or sunshine	washing yourself by standing upright under water sprayed from a nozzle
the act of washing yourself (or another person)	a plumbing fixture that sprays water over you
	a sudden downpour (as of tears or sparks etc) likened to a rain shower
	a party of friends assembled to present gifts (usually of a specified kind) to a person
	a brief period of precipitation
	someone who organizes an exhibit for others to see

bathing (verb)	shower (verb)
	take a shower; wash one's body in the shower
	provide abundantly with
	spray or sprinkle with
	expend profusely; also used with abstract nouns
	rain abundantly