Ms. Musharbash, 21 years old, was surprised he hadn’t already figured it out. Soon after meeting him on the dating app Tinder, she discovered his last name through his Snapchat screen name. But his question meant things were getting serious.

“He now knew me as a whole person,” she said.

As online dating has proliferated, so too have an array of norms that might seem bizarre—or downright counterproductive—to generations who didn’t rely on their phones as a way to meet people. Among them: a reluctance to ask for surnames until the relationship has progressed to a more serious level.

Asking for a last name “is definitely a modern social cue” that trust is building in a relationship, said Denny Dowty, a 26-year-old in Kansas City, Kan. “It’s the 21st-century equivalent of leaving a calling card.”

Many millennials say asking directly for a last name on a first date feels awkward, and signals too obviously they intend to scour the internet for biographical information. Others say that downloading a date’s entire digital footprint—armed with the full name—can stop a relationship from developing organically.

“The less I know, the better,” said Brendan Krick, 25, a comedian in Philadelphia. “Everyone is so lame on the internet.” Just seeing that a woman liked “a bunch of bands that suck” on her Facebook page could be a deal breaker, he said.

Dating apps have grappled with how much information to reveal about a user without sacrificing safety. Tinder and Bumble, another service, typically show only first names. OkCupid recently revised its policy to require people to use their first names instead of made-up usernames such as “DaddyzPrincess29.”

The League, another app, generally requires users to show their first name, last initial, educational background and current employer, saying “you’ll never have to wonder if that Harvard hottie is too good to be true.” Users can pay extra to hide their employer, or customize in other ways.

Nicole Ellison, a University of Michigan professor who has studied online dating, said finding out last names can shatter the carefully curated image presented through an app, potentially revealing what people are actually like.

“Once you have the last name, that unlocks this whole new universe of information,” said Prof. Ellison. “You can go to their social media sites, Google the person, look up criminal histories.”

Thanks to sites such as LinkedIn and Facebook, an amateur sleuth can often figure out last names with just a first name and the person’s alma mater or place of work.

When that doesn’t work, there are other tactics, such as taking a peek at an Uber account name or credit card after a date, or asking to exchange social media handles.

Hayden Moll, a freshman at Missouri State University, swiped left on Tinder recently on the profile for a woman named Claudia, even though he meant to swipe right to show his interest. Undeterred by the mistake, Mr. Moll saw they attended the same school and proceeded to email 42 different Claudias at Missouri State hoping to find her.

“If Tinder provided last names this would be much easier,” he wrote in the email.

Mr. Moll said the two plan to meet up for doughnuts “sometime soon.” Ms. Alley didn’t respond to a request for comment.

Dating app users such as Judith Cothran say they don’t want to put in the effort to learn last names until relationships become more serious. Ms. Cothran, 41, who works in software sales, said she has been on more than 300 first dates since 2010, but has only learned the last names for approximately 20 of them.

“I realized that pre-date research is often a waste of time,” she said, especially because it isn’t guaranteed to reveal whether the man is married, her biggest concern.

Mike Ingram, a 41-year-old books editor, said it was easier to find out last names when he started online dating about a decade ago because people commonly exchanged email addresses, which often contained last names, instead of phone numbers. “Nobody emails anymore,” he said.

Keeping track of a one-name first-date phone number isn’t easy. Mr. Krick said his phone contacts contained, at one point, more than 60 women with the last name “OkCupid” or “Tinder.” He hardly ever enters real last names.

“The one time I did it recently, it was because there was another Emily Tinder already, so I needed to find out her real last name,” he said. “I couldn’t have Emily Tinder Two.”

Jessica Lieberman, a 25-year-old user-experience designer in Washington, D.C., said she puts prospective suitors in her phone using nicknames, such as “Dave Military” or “East Coast Science Dan.”

Changing a last name in her phone from “Tinder” to a real last name is a “modern relationship milestone,” said Angelica Guarino, a 20-year-old student at Boston University.

Some online daters thoroughly investigate people before agreeing to meet, including copying and pasting photos into Google’s image search.

Maria Mir, 19, said if she has only a first name and location for a prospective date, she deploys friends to find him on Facebook or other social media sites, partly to check out his political views.

“I want to know what you think about mass incarceration and things like that,” she said.

This can also backfire. A man Ms. Lieberman matched with on Bumble told her, on the second date, he had found her LinkedIn page. He brought up her education status and mentioned a nonprofit job she held around 2014, none of which she had yet discussed with him.

Ms. Lieberman said she judged him for scrolling through her LinkedIn.

“That ended terribly,” she said. “It wasn’t a match.”

Write to Nicole Hong at nicole.hong@wsj.com