The Irish 'Blackbird' air/set dance is an allegory for Bonnie Prince Charlie, not so much the bird.

But there are tunes like the 'Morning Thrush' which James Ennis composed based on the song of a thrush in the ivy outside his window. His son Séamus made a nice job of playing it.

There is also the recording 'The Grouse in the Heather' by PJ and Marcus Hernon that is a collection of tunes they composed :

The Grouse In The Heather
The Hunted Pheasant
The Snipe In The Marsh
The Curlew’s Cry
The Invisible Corncrake
The Peeping Plover
The Squeaking Woodcock
The Chattering Stormcock
The Musicial Thrush
The Herring Gull
The Diving Gannet
The Golden Plover
The Bobbing Sandpiper
The Linnet’s Chorus
The Beautifull Goldfinch
The Hovering Kestrel
The Lady’s Falcon
The Lonely Bittern
The Nesting Goldcrest
The Warbling Robin
The Kingfisher’s Delight
The Dark-eyed Raven
The Barefaced Crow

There's a bird theme there I think.

Tunes like The Gander in the Pratie Hole, The Geese in the Bog (and the Dogs among the Bushes) have an arguably an optional element of onamonapia when played on the pipes.