Elon Musk has long been a figure whose ambition often seems boundless. Yet, his latest endeavor — a goal to push Tesla’s market value to a staggering €1.9 trillion — is setting the stage for a new chapter in both the company’s history and the wider tech world.

Mr. Musk says the driverless taxis could begin ferrying passengers on Sunday in Austin, Texas, where other companies already have similar cars on the road.