Does each stable in a pasture double the number of animals allowed?
Each stable doubles the capacity of the pasture.

While not usually a great move, there are situations where this could be useful.

From the rules, top right on page 9: (Emphasis is mine)


  Each stable costs 2 Wood and can be placed on a fenced or empty unfenced farmyard space. 
  A fenced stable doubles the holding capacity of a pasture. Each farmyard space can hold 1 
  stable. Each space within a pasture may have its own stable: the pasture’s capacity is then 
  multiplied by 4, 8, etc.