Jake Kassan, 26, likes to play a little game: ask other 20-somethings for the time, and see if they glance at their wrists or their phones. “More often than not, they reach for their phone,” he said. This might seem like a problem, given that Mr. Kassan and his business partner, Kramer LaPlante, also 26, run Mvmt, a watch company that targets millennials.

But Mr. Kassan could care less. “Watches have evolved,” he said. “Our audience cares more about the style of a watch than its function.”

Millennials are often thought to be the lost generation when it comes to watches, since they were raised with cellphones and, according to cliché anyway, are often too busy struggling to carve out careers in an uncertain economy to fritter away money on their yuppie parents’ status symbols.

Funny, then, that a Los Angeles-based watch start-up founded by a couple of college dropouts in 2013 has become an industry player (the company says its revenues were $60 million last year) by selling old-school timepieces — dumb watches, if you will — to people too young to remember rotary phones.