PORTSMOUTH, N.H.—Then-Sen. Barack Obama had to spend precious days at the close of his 2008 presidential campaign to defend against charges of socialism after he casually suggested his tax plan would “spread the wealth around.”

Today, seeking the same job in the same party, Sen. Bernie Sanders proudly wears the democratic socialist label and touts a sweeping plan designed to spread the wealth. He calls for a “political revolution” to vastly expand the government’s role in American life and sharply cut the power of Wall Street...