The problem is that families don't, at least within the meaning implied by the above statement. Personal income in the United States is about $12.9 trillion. Mortgage debt outstanding is about $10.5 trillion. Consumer credit outstanding is about $2.4 trillion. So the ratio of consumer debt relative to consumer income is similar to US government debt to GDP.
I am not saying we needn't worry about long term fiscal balance--we do. As I have said before, we must, among other things, return tax revenues to at least their long-term mean as a share of GDP, and bend the cost-curve for health care--something that Obamacare actually tries to do. But bad metaphors that basically dishonestly flatter people are not helpful.