When Bob Schneider goes out on a Friday night, he no longer heads to the local singles bar. Instead, he goes to the grocery store.

“I once dated a woman from the potato section at Mariano’s,” says Mr. Schneider, a 67-year-old semiretired, twice-divorced lobbyist in Oak Brook, Ill. “The next thing you know we’re at the wine bar and then we’re dating.”

Supermarkets—those...