It handled more payments last year than Mastercard, controls the world’s largest money-market fund and has made loans to tens of millions of people. Its online payments platform completed more than $8 trillion of transactions last year—the equivalent of more than twice Germany’s gross domestic product.

Ant Financial Services Group, founded by Chinese billionaire Jack Ma, has become the world’s biggest financial-technology firm, driving innovations that let people use their phones for buying insurance as easily as groceries,...