AN OLD WORKPLACE adage says you shouldn’t wear a watch that’s fancier than your boss’s. For Tony Traina, 27, an associate lawyer in Chicago, however, it would have been difficult to find a more casual timepiece than the $399 Apple Watches his firm’s higher-ups tend to favor. He didn’t bother. As a watch-lover, he ignored protocol, showing up at work in a $1,500 mechanical watch by German brand Nomos Glashütte. But it barely went noticed.

In the past, one’s shoes, one’s suits and, yes, one’s watch spoke volumes about one’s...