jmc

The positive multiples of $63$ are $63$, $126$, $189$, .... All of these numbers except for $63$ are bigger than $63.$ On the other hand, all the positive factors of $63$ other than $63$ are less than $63.$ So if Marie and Jay are thinking of the same positive number, that number must be $63.$

By the same reasoning, the list $-63$, $-126$, $-189$, ..., of negative multiples cannot include any factors of $63$ except $-63$. Thus the two possible values for the number Marie and Jay are thinking of are $-63$ and $63.$ The product of these two numbers is $(-63)(63)=\boxed{-3969}$, since the product of a positive number and negative number is negative.