However, when the recipient logs in and looks at his/her messages, the sender's (formatted) name links to their profile, which we don't want. Is there a way I can disable this link? Some setting which is similar to the one in User Lists where you can choose not to link to User Profiles?

That's the userlist display of the formatname field. It's used in a lot of our plugins and locations. So the best way to probably get rid of its linking is to just override its userlist output entirely. To do this edit the formatname field within CB > Field Management then under Parameters > Layouts adjust "Userlist Value Layout" as needed to output whatever you like. For example if you want it to be their username set it to [username] and if you want it to be their name set it to [name].