1. What does it mean?
댕댕이 is a visual wordplay spelling of 멍멍이, an affectionate word for a dog. The joke depends on the resemblance between written Korean letter shapes.
Also written or said as 멍멍이, daengdaengi, doggo
2. The nuance
It is not the formal name of a breed and is not limited to puppies. It suits a pet caption or a friendly reaction to a dog photo. For a factual notice or a veterinary conversation, 강아지 or 개 is clearer.
3. Examples
“산책 나온 댕댕이가 낙엽을 한참 구경했어.”
A doggy out for a walk spent ages looking at the fallen leaves.
“우리 집 댕댕이는 공보다 작은 인형을 더 좋아해.”
Our pup likes a little stuffed toy more than a ball.
Sources
National Institute of Korean Language
References support the meaning and usage. Example sentences are written for NoCapWiki.