Susan E. Rice, President Barack Obama’s last national security adviser, called the arrangement “stone cold crazy” in a tweet posted Sunday.

Mr. Spicer said the language the Trump White House used in its N.S.C. executive order is, with the exception of Mr. Bannon’s position — which was created during the transition — almost identical in content to one the Bush administration drafted in 2001. And Mr. Obama’s top political operative, David Axelrod, sat in on some N.S.C. meetings, he added.

There were key differences. Mr. Axelrod never served as a permanent member as Mr. Bannon will now, though he sat in on some critical meetings, especially as Mr. Obama debated strategy in Afghanistan and Pakistan. “It’s a profound shift,” Mr. Axelrod said. “I don’t know what his bona fides are to be the principal foreign policy adviser to the president.”

But Mr. Bannon’s elevation does not merely reflect his growing influence on national security. It is emblematic of Mr. Trump’s trust on a range of political and ideological issues.

During the campaign, the sly and provocative Mr. Bannon played a paradoxical role — calming the easily agitated candidate during his frequent rough patches and egging him on when he felt Mr. Trump needed to fire up the white working-class base. The president respects Mr. Bannon because he is independently wealthy and therefore does not need the job, and both men ascribe to a shoot-the-prisoners credo when put on the defensive, according to the former Trump campaign manager Corey Lewandowski.

Mr. Bannon is a deft operator within the White House, and he has been praised by Republicans who view him skeptically as the most knowledgeable on policy around the president. But his stated preference for blowing things up — as opposed to putting them back together — may not translate to his new role.

The hasty drafting of the immigration order, and its scattershot execution, brought a measure of Mr. Bannon’s chaotic and hyperaggressive political style to the more predictable administration of the federal government. Within hours of the edict, airport customs and border agents were detaining or blocking dozens of migrant families, some of whom had permanent resident status, until John F. Kelly, the new homeland security secretary, intervened.