Was there anything special about this bird, other than it being Ivan's pet?
Ivan is, apparently, attached to the original bird but even when Hammer gets him an obvious substitute Ivan seems semi-content. At no point does he STOP working (apparently) for Hammer*
Frankly, Ivan Vanko has been in a Siberian gulag for many years...it's unlikely the bird is even alive.
Justin: Hey, hey, hey. Special delivery. Candygram. [Ivan climbs down from where he was working] I brought you something. Oh yeah. [Removes cloth] We got you the bird, pal.
Ivan: This is not my bird.
Justin: What do you mean? That’s the bird. This is the bird. Yeah, pulled a lot of strings to get this bird. This is a great bird.
Jack: It’s a beautiful bird.
Justin: We got this bird all the way from Russia.
Ivan: Hey, man, this is not my bird.
Justin: Well, listen, even if it’s not the bird, I mean, this is a gorgeous bird.
When Hammer takes the bird away, Ivan just shrugs it off pretty much...
Justin: Hey, there he is. It’s the bird man. Now you like the bird. Is that right? Is that your bird? I’m confused. You said it wasn’t, but now it looks like you’re the best of pals. You love that bird, don’t you? You know what? Take the bird.
Ivan: [As man grabs bird] Hey!
*Of course, it's pretty obvious that Ivan was always planning on betraying Hammer as was just using him for his own ends.
To that extent, regardless of his affection of the original bird, it was unimportant in light of his desire for revenge on the Stark familiy