I was setting up a new iPhone today and had the same issue. I was able to resolve it as follows:
I logged in to my Apple ID on the web (via Eric's link, above). In the "Account" section, there's a subsection called "Reachable At". I had three items there: the phone number for my iPhone, the email address for my Apple ID, and a second email address that I'd added at some point. I clicked "Edit" and deleted the second email address. Once that was done, I could select Settings > Messages > Share Name and Photo without getting the "multiple phone numbers" error.
My hypothesis is that it's a poorly-worded error message. It may mean to say that you have multiple phone numbers and/or email addresses associated with your Apple ID, beyond your iPhone's phone number and your one Apple ID email.
Hope this helps.
-Ao