dressing case vs baggage

dressing case and baggage both are nouns.

Word	Noun	Adjective	Verb	Adverb
dressing case	Yes	No	No	No
baggage	Yes	No	No	No

As nouns, baggage is a hypernym of dressing case; that is, baggage is a word with a broader meaning than dressing case:

dressing case: a small piece of luggage for carrying brushes and bottles and toilet articles while traveling
baggage: cases used to carry belongings when traveling

Other hypernyms of dressing case include luggage.

dressing case (noun)	baggage (noun)
a small piece of luggage for carrying brushes and bottles and toilet articles while traveling	cases used to carry belongings when traveling
	the portable equipment and supplies of an army
	a worthless or immoral woman