The former Manchester United player was hired last week, but the Major League Soccer club mistakenly used a photo of Ajax youth manager Tinus van Teunenbroek to welcome Stam.
The club then deleted the tweet and reposted with the correct image of Stam.
"I was quite surprised, to be honest," Stam said. "I got the picture over here from a friend in Holland and I thought he made a practical joke by orchestrating the picture himself.
"Afterward, I got a message from somebody else as well, so after that, I was quite curious what was going on. So I saw the picture on the website and I was quite surprised, but mistakes are made and I can laugh about it as well."
"As a head coach, you're always looking for that special job where you can build something, and I think FC Cincinnati has already done great over the last couple of years," he said.
"They're building on the team, on the training ground, facilities, new stadium and, of course, the prospect of going to the top of MLS. That's a great project for myself as a manager."