Other mistake: Right at the beginning of the movie, when Lewis's real mum picks him up from the box and hugs him, it's no more than three seconds before she hears Lewis from the future slipping on the step. She gets scared and puts baby Lewis back in the box. When Wilbur takes Lewis to meet his mum, near the end of the movie, the time his mum holds the baby is more than twenty five seconds.
Other mistake: In the beginning, when Goob falls asleep, he misses the catch, by almost a foot or so. At the end of the movie when Lewis calls out to Goob, the ball lands on his head, and drops into his hand. Since Lewis didn't alter the timeline to where Goob is standing, (as to the part of not inventing the helping hat) Goop should have been in the same spot.
Other mistake: In the beginning of the film when Lewis is dropped off at the orphanage there is a blue car parked in the background. This is the same exact car (license plate and all) that he is driven away in when adopted at the end.
Answer: There's no indication that the dog had anything to do with Mr Peabody (especially since the dog didn't speak or do anything other than sit there). It was just a front for a throwaway joke.
It's a coincidence because Mr. Peabody and Sherman was not released until 2014 and Meet the Robinsons came out in 2007.
But that film is based on the Peabody's Improbable History segment from the 50s/60s TV show The Adventures of Rocky and Bullwinkle and Friends.