Donald Trump Jr., the president’s eldest son, doesn’t always take after his father. He finds little joy in wandering gilded rooms. He would rather go hunting than play a round of golf. He has little interest in becoming a politician.

He has, however, inherited his father’s affection for Twitter as a weapon against political foes. But even here, their styles are different.

President Trump tends to fire a digital bazooka when met with a perceived slight, often hitting below the belt and leaving himself open to bipartisan criticism, as he did this week by insulting the hosts of the MSNBC program “Morning Joe.” The younger Mr. Trump uses the platform more frequently to harshly call out what he sees as a bias against the Trump White House that is shared by journalists, Democrats and celebrities — all suspects, he feels, in a plot to undermine and threaten his father’s embattled presidency.

Some of the president’s supporters privately say Mr. Trump is doing a more forceful job defending his father from outside the White House than some people inside it, and some see a widening opportunity for him to be an effective and high-profile torchbearer for their grievances.