What does the idiom in My Dog mean?

It refers to crows.

In China culture, crow is usually regarded as an ominous sign, but in the idiom "Love me, love my dog", it symbolizes respect and love for the people I love. This idiom means a person's admiration for the object he loves, not only the object itself, but also the people or things related to it. Just like love for a house, it will extend to love for crows in this house. This is because respect and love for loved ones make people have the same feelings for people or things related to them.