From the Super Mario Wiki
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This cat is the family pet of Junior and his parents. Due to the fact that it has to live in a house with the out-of-control Junior, the cat appears to have suffered some degree of abuse, as evidenced by the bandages on its leg and forehead.
When Junior broke two vases, he scrambled to place the cat on the broken pieces, tricking his parents into thinking that it was the one who broke it. As a result, the mother scolded the cat.