How did Jack get her name?

If you look under the heading "goofs", that is clearly stated. It was a mistake. I think the dog was just supposed to be a cute link between them that really had no other purpose, and as such, was not scrutinized for errors too much by directors and producers

Actually Kate owned the dog first (2004), but the story leaves a void on what happened to the dog for a time period when she was first dating Morgan. Note that Morgan did not recognize the dog. In most scenes the dog is time traveling back and forth in time between Kate and Alex. Note that the dog was never with both of them simultaneously in any scene in the movie.

No explanation was provided on the reasoning for the name beyond Alex noticing that the dog answered to Jack after Kate mentioned it in the letter.