1. What does it mean?
Short for 별것을 다 줄인다, a reaction to an unfamiliar or surprisingly elaborate abbreviation. Its joke is self-referential: the complaint about shortening every little thing has itself been shortened.
2. The nuance
It can sound amused, baffled, or mildly exasperated. It is not the name of a particular slang trend, and it does not necessarily mean the speaker dislikes the person using the abbreviation.
3. Examples
“많관부가 많은 관심 부탁드립니다라고? 진짜 별다줄이네.”
Man-gwan-bu means 'please show lots of interest'? They really do abbreviate everything.
“새 줄임말을 볼 때마다 별다줄이라는 말부터 나와.”
Whenever I see a new abbreviation, my first reaction is 'they shorten absolutely everything.'
Sources
National Institute of Korean Language
References support the meaning and usage. Example sentences are written for NoCapWiki.