Was there something special about Ivan's bird?

Was there anything special about this bird, other than it being Ivan's pet?

No...

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

To add more info to already good answer, and answer a comment question by OP:

Any idea of why Ivan cared about the bird so much? I was wondering/thinking maybe it had to do with his father or sth..

From Quora:

Actually, the bird was Mickey Rourke's idea. It wasn't in any of the drafts of the script, but Rourke wanted something to establish that Ivan was not a terrible guy; he just had some serious revenge issues. He wanted to show that he was able to form an emotional bond (apparently the bond with his father in that opening scene just wasn't strong enough for him).

This is a round about way of saying that, when Ivan gets the second bird, he's upset that Hammer didn't actually get his bird from Russia. This does not mean that he can't bond with the new bird. My thought is that he makes the most of it, and makes friends with the bird they've brought. It isn't the bird's fault, so why take it out on the bird? Ivan's emotional side (that Rourke tries so hard to establish) dictates that he also likes this bird, too.

also:

When he gets to Hammer and asks for his bird, he is testing Hammer to see how far he can push him, knowing already what he wants to do… Hammer needed him, so he agreed to “the bird”, which became a bird - much the same as suits became drones… Hammer thought he was being clever but Ivan had already sussed him and was 3 steps ahead.

To add a source if you don't believe the guys from Quora:

According to the actor, the bird wasn't the only aspect of his character he came up with and pushed for in the film.
"I bounced these ideas off [Jon] Favreau. He had to fight for me to be okay with the teeth and tattoos and the bird," said Rourke. "Everything was a little bit of a fight, but once I justified it, Favreau had to go to the powers that be."