This can indicate restraint or entrapment or feeling stuck in a job, lifestyle or relationship. Is your freedom somehow restricted in waking life? Are your inhibitions stopping you expressing your personality? Is your job humdrum? Is your partner possessive? If you opened the door in your dream and let the bird out to fly away with joy in your heart, this is almost certainly a message from your unconscious to take steps in waking life to release yourself from restrictions and or inhibitions.
If the bird wouldn’t or couldn’t fly away, are you putting limitations on yourself? If the bird’s wings were clipped, is someone deliberately holding you back? Freeing a bird from its cage relates to expressing your emotions.
If the bird in the cage is a pet bird, this often suggests happy memories. See also PETS.... The Element Encyclopedia
To see wild animals caged suggests that you are in control of your instincts.
If you are in the cage with them, it could suggest a need to break free from constraints.
If you dream of an animal tangled in barbed wire or in a trap, this could suggest an unhappy relationship with yourself or someone else.
If the animal seems calm, it suggests inner strength during adversity but if the animal panics, unhappy memories or unhealthy habits are limiting your potential for development and growth.... The Element Encyclopedia