The old shoe is a reminder of the Monopoly Man's past. It once belonged to a young orphan, who every day saw the businessmen and women walking in their crisps suits down the street, and thought, "One day, I too will be great like those people."
Every time he sees that shoe, the Monopoly Man smiles and remembers the look on that boy's face as his orphanage was bulldozed for a golf course, shortly before the terrier maimed the child for life.
Also, how could Cam forget the top hat? It's the most stylish of pieces.