嗜み means two different things:
- hobby; enjoyment as an amateur
- what you should know to be a refined X; basic knowledge/experience for X
So 大人の嗜み can refer to either:
- hobbies only adults can enjoy (e.g., wine and cigarette. See this article for example)
- what you should know/have as a mature adult (See this article for example)
メイドの嗜み almost certainly refers to "what you should be able to do as a maid" because "hobby (only) for maids" doesn't make much sense to me.
The verb version, 嗜む, also means two things, "to casually enjoy something as a hobby" and "to have the basic knowledge/experience on something".
嗜む/嗜み is an elegant word that is typically used with uncommon "upper-class" activities like 乗馬, テニス, チェス, ピアノ and 生け花. Saying アニメを嗜む is not wrong but may sound a little funny.