What I remember:
Blaine was a stray (hybrid dog with human traits), and while out he found a collar/other pet with a collar and was desperate to put it on and feel wanted, so he did but he got caught by the pound. They called Kurt who thought it was his actual pet who had gone missing (I think he was named after a designer?) and was kind of bitchy/mean when he found out it was Blaine instead.
EDIT: I think that Kurt's first pet was named "McQueen"? He escaped from the yard after a noise scared him or something. Dogcatchers find Blaine who has put on the collar and call Kurt to tell him that they found McQueen. For the first part of the fic Kurt is pretty mean to Blaine. I think at one point Blaine runs away?
I also may have been mistaken about the human/dog hybrid, he may just be a human!pet (they were bred specifically to be a pet)
Eventually Klaine, but the family accepts them and they have to keep it a secret.
EDIT: FOUND! Link in comments :)