hair coloring vs hair dye

hair coloring and hair dye both are nouns.

Word	Noun	Adjective	Verb	Adverb
hair coloring	Yes	No	No	No
hair dye	Yes	No	No	No

As nouns, hair coloring and hair dye are synonyms defined as:

Other synonyms of hair coloring include tint.

hair coloring (noun)	hair dye (noun)
the act of dyeing or tinting one's hair	a dye for coloring the hair
coloring of the hair
a dye for coloring the hair