Ok so I head canon that each of bruces boys has a different bird name they answer due to it being confusing to say “robin” since all 5 used the "Robin” title at one point (though Duke only ever used it unofficially)
Dick Greyson, Nightwing, is Bluebird - Blue birds are blue like his costume and often associated with cheerfulness and hope.
Jason Todd, Red Hood, is The Cardinal - Because its red. Enough Said.
Tim Drake is Oriole - I don’t like Red Robin purely because its too confusing with both Red Hood AND Robin. Tim is associated with both red and yellow but since Jason has Red and Duke has yellow for Tim I figured why not mix the two together and give him orange, and what bird is orange? Orioles. Also orioles are a top tier underrated bird.
Duke Thomas - The signal - Gold Finch since his costume is yellow and Dinah already has canary.
Damien Wayne - is the current Robin but given his association with green His brothers Jokingly call him Parakeet. (he hates it)
Bruce is sometimes jokingly referred to as The Crow (he aso hates it)
and Alfred is on occaision referred to as magpie
I dont know enough about steph or cass to speculate as to their bird names but you are free to if you want. Maybe Black Dove for Cass? IDK