Interestingly there are surprisingly few pictures of the bird exhibits at Cologne, even though there are plenty of them. By going through Zootierliste it appears there aren't too many exhibits in Cologne with true island / Oceania species. There are still quite a number around, but the advantage is smaller than I first thought.
The bulwark of birds at Cologne is the Pheasantry and here there are some 4 aviaries with relevant species. 2 small, but well-designed Australian aviaries house several smaller parrots, some obligatory ducks as well as bush thick-knee and masked lapwing. Noteworthy is that Cologne was the first European zoo in recent decades to import wild-type budgerigars, which have started spreading since. There are no pictures of these aviaries around. Additionally there is a Malagasy aviary with crested ibis, Madagascar turtle dove, fody & blue coua (and some non-Malagasy birds IIRC). Also no picture, but it is similar in size to this one:
@Toddy
The former hippo enclosure is now an aviary with among others a selection of ducks that occur in Madagascar (and mainland Africa), Malagasy sacred ibis, Madagascar pond heron (and on my most recent visit also milky stork):
@lintworm - vegetation has grown since this picture was taken, so it looks more attractive now.
The wonderful S-American house has Cuban ground doves, crested quail-dove & soccoro pigeon (and plenty of other birds & primates):
@twilighter
A large variety of Oceanian waterfowl species is kept on one large pond, with more special species like NZ scaup in side enclosures. Other ponds house species that also occur in the Caribbean (either regular or only as winter visitors) or Indonesia.
@Hvedekorn
Normally the rainforest house would be the place to see a large variety of Indonesian bird species (there used to be some 30 species on show IIRC), but after the recent fire & first renovation and the coming renovation, I am not sure how many species are actually on show. For sure most of the rarest pigeons & doves as well as the birds-of-paradise remain off-show. This is a picture from before the fire:
@lintworm
Burgers' Zoo clearly has less island birds on-show, but there is still quite a variety including 2 that are unique for Europe and plenty of other rarely kept species, often with good breeding results. The Bush holds the following:
Madagascar crested ibis
Victoria crowned pigeon
Green naped pheasant pigeon
Pinon imperial pigeon
Green imperial pigeon
Luzon bleeding heart pigeon
Grey-capped emerald dove
Nicobar pigeon
Superb fruit dove
Crested quail dove
White-rumped shama
Chestnut-backed ground trush
Fairy bluebird
Scissor-billed starling
Montserrat oriole
Madagascar red fody
Red-whiskered bulbul
White-eared catbird
The Desert has:
Socorro pigeon
Blue-winged teal
Painted bunting
Common ground dove
And the Mangrove has:
Red-legged honeycreeper
Indigo bunting
For each zoo I have excluded species that are basically mainland species, but also occur on a few (small) islands, like Trinidad, Zanzibar or Great Britain. Including those would mean that almost any species becomes an island species, especially when it comes to birds.
Overall Cologne outnumbers Burgers' in birds in something like 3-1, not counting species only held off-shwo. but a sizable part of the collection advantage are pinioned waterfowl, from an exhibit standpoint, Burgers' remains superior, though the average level of aviaries in Cologne is good too.