The same week President Donald Trump was sworn into office, he applied for a trademark on “Keep America Great” for use on buttons, banners, clothing, advertising and more, according to the U.S. Patent and Trademark Office, and filed the paperwork to launch a re-election campaign earlier than any other president in modern history.

Mr. Trump’s early political moves stand in contrast to that of President Ronald Reagan, who said deep into his third year in office, “It is far too early for anyone of my position to announce his...