"My dog goes with me."
Translation:Mijn hond gaat met me mee.
Maybe if you added a direction. "De hond gaat met me naar het strand" (the dog is going with me to the beach) is fine. "Meegaan" means "to come along" so without "mee", the focus is less on the person coming along.
"Mijn hond gaat met me" on its own doesn't work. I think it has a similar meaning as "to be seeing someone"; to be in a relationship with someone.
Good question! Maybe they wanted an exercise with the verb "to go along" (meegaan) concretely, and it just happens that you can (this is only my hypothesis) skip the verb "meegaan" and say "gaan bij me" instead. It's normal in a language to have more than one way of saying things.