I went back to read chapter 46, and yup there was never a definitive conclusion to Azi’s story, but they brought back the owner of the dog that died, so I get the feeling that he kept Azi.
I just feel like it was too much of a coincidence that they said they would look for someone for Azi and that character comes back to talk to Jun.
At least that’s my head canon. Azi was adopted by that character.

Whatever happened to Azi? The vet was like I'll watch over him until he recovers, " and then Azi just disappears (chapter46). I spent the next 80 chapters waiting for an update on Azi. We never got to see Azi happy, I thought he'd adopt Azi but he adopted a different dog.