Kima Nieves recently received two Aveeno bath-time sets and a box of Huggies diapers through her baby registry on Amazon. The only problem? The new mother didn’t ask for the products, or even want them.

Instead, Johnson & Johnson and Kimberly-Clark Corp. each paid Amazon.com Inc. hefty sums to place those sponsored products onto Ms. Nieves’s and other consumers’ baby registries. The ads look identical to the rest of the listed products in the registry, except for a small gray “Sponsored” tag. Unsuspecting friends and family clicked on the ads and purchased the items, assuming Ms. Nieves had chosen them.

“Very sneaky,” said the 28-year-old health-care analyst from Fredericksburg, Va. “That’s friends’ and family’s money going somewhere we didn’t approve of.”

Amazon in recent years has charged into advertising, building the third-largest digital ad business in the U.S. after Alphabet Inc.’s Google and Facebook Inc., according to eMarketer. Its ad revenue is on pace to double this year, to $5.8 billion, eMarketer estimates.

As Amazon has monetized more space on its website, shoppers are increasingly encountering sponsored ads. Amazon, for instance, enables advertisers to pepper the tops of search results with sponsored products that look similar to regular listings, save for a “sponsored” tag. Within product pages, brands can choose to promote other products and customize the title, such as “Top Men’s Running Shoes.”