hair dye vs hair coloring

hair dye and hair coloring both are nouns.

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

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

Other synonyms of hair dye include tint.

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