dressing case vs luggage

dressing case and luggage both are nouns.

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

As nouns, luggage is a hypernym of dressing case; that is, luggage 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
luggage: cases used to carry belongings when traveling

Other hypernyms of dressing case include baggage.

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