Historians (UK) prefer dovecotes to name the tall buildings of brick or stone (or even timber and plaster) with nesting holes for hundreds of birds. Pigeons lay eggs for most of the Spring and Summer so eggs and squabs provided food. But the main economic reason was the dung, high in nitrogen and phosphates,
Dovecote today is the word for a garden ornament. The doves will usually be white doves or fantails.
Welsh Dovecotes - Wall mounted Dovecote for sale
Those who keep tumblers to watch their flight and hold aerobatic competitions may simply call their home a box, or the box
If the birds are kept for their plumage, they will usually be housed in an aviary. This photo accompanied an advertisement: "selling due to lose of mate + i need to remove the aviary."
Loft or Pigeon Loft is the word used by most pigeon fanciers especially those who keep racing- / homing-pigeons, even if they are housed in a shed at ground-level.