Manger:
* Latin: *manducare* (to eat)
* Old French: *mangiere* (trough for feeding animals)
* Middle English: *manger*
Stable:
* Latin: *stabulum* (stall, stable)
* Old French: *estable* (stable)
* Middle English: *stable*
So, while the words themselves have evolved over time, their core meanings relate to the fundamental functions of a manger (feeding animals) and a stable (housing animals).