The character name only shows up if they are playing WoW and on a character from your realm and faction and it has click interaction and is colored by the toon's class.
The addon also adds a tooltip to the [Battle.net Friend Name] link which has their toon name, realm (if different from yours), faction, level, race, class and game client ID. The tooltip is displayed even if they are on a different game/realm/faction.
I haven't yet tested the tooltip feature with clients logged into multiple sub-accounts of one battle.net account, battle.net accounts not logged into a game client (logged into the battle.net website for example) or clients logged into other battle.net games. If you run into any issues with one of these scenarios, please, let me know as much as you can about the error or strange addon behavior you experienced and the scenario that caused it.
Another note on the tooltip function: if you use any third party chat frame mods which completely replace the default chat frame(s) the tooltip functionality probably will not work, but if you let me know which chat mod you use then I'll try to include support for the more popular ones.
Any chat mods which use ChatFrame, ChatFrame1, ChatFrame2, ChatFrame3, etc should work fine, but if they create their own chat frame and hide the default frame(s) and don't use one of those names for their frame then chances are really good I'll have to add support for it specifically.