The unique characteristics of the digital world, including the inability for both consumers and regulators to see the algorithms that determine what users see in search results and news feeds, and what advertisers pay to reach them, make righting any potential wrongs in the industry even harder, Mr. Kimmelman, who now works for consumer-rights nonprofit Public Knowledge, said.

Ms. Vestager’s dual role as both a setter and enforcer of European Union rules could raise concerns about conflicts of interest and investing too much power in one office’s hands.

As part of her antitrust portfolio, she will be responsible for assessing if companies are using their market share to unfairly block out smaller competitors. And from her new perch as digital regulator, she can advance new rules on company behavior, including how they collect, use and store users’ data; and how they shield themselves against cyberattacks.

“Concentration of power is an issue,” said Mr. Vinje, the antitrust lawyer. “There is an efficiency benefit in having those powers combined in one person, but on the other hand there is always a benefit in debate and division of power, so one can question if this results in too much power.”

In her regulatory role, Ms. Vestager will be able to propose new legislation, but any law will need to be approved by European Union member states and the European Parliament, meaning her powers will be closely checked.

Ms. Vestager will also continue to oversee ongoing investigations, including whether Amazon is unfairly using data from third-party sellers who use its platform to improve its own offerings. Her office has also faced pressure to investigate if Apple is unfairly using its app store to harm competition, and to initiate further inquiry of Google.