1. What does it mean?
Short for 취향 존중, 'respect for personal taste.' It is used to ask people to leave room for different preferences in music, food, clothing, hobbies, or other matters of taste.
Also written or said as 취향 존중
2. The nuance
취존해 줘 means something like 'let me like what I like.' Among friends it can be playful, but it can also set a boundary against persistent teasing. Respecting a preference does not require sharing it.
3. Examples
“나는 이 결말이 좋았어. 취존해 줘.”
I liked that ending. Let me have my own taste.
“음악 취향은 달라도 서로 취존은 해야지.”
Even if our music tastes differ, we should respect each other's preferences.
Sources
Aju Business Daily: Korean expressions
References support the meaning and usage. Example sentences are written for NoCapWiki.