birdcage vs cage

birdcage and cage both are nouns.

Word	Noun	Adjective	Verb	Adverb
birdcage	Yes	No	No	No
cage	Yes	No	Yes	No

As nouns, cage is a hypernym of birdcage; that is, cage is a word with a broader meaning than birdcage:

birdcage: a cage in which a bird can be kept
cage: an enclosure made or wire or metal bars in which birds or animals can be kept

Other hypernyms of birdcage include coop.

birdcage (noun)	cage (noun)
a cage in which a bird can be kept	an enclosure made or wire or metal bars in which birds or animals can be kept
	the net that is the goal in ice hockey
	a movable screen placed behind home base to catch balls during batting practice
	something that restricts freedom as a cage restricts movement

birdcage (verb)	cage (verb)
	confine in a cage