When T’Challa, Shuri, Okoye, and Ross are looking at the hologram of Killmonger, they mention that he has “a War Dog tattoo” but that they don’t have him in their records. (This is right before Ross says “he’s one of ours.”) Therefore, it seems reasonable to assume this tattoo is specific to War Dogs, although Killmonger only has one presumably because his dad inscribed it on him before his death as a form of identification should he ever go to Wakanda.
Note that T’Chaka is never shown to have this tattoo.
The War Dogs, contrary to what the above linked question/article imply, don't seem to be a tribe of their own. There are five tribes (River, Border, Merchant, Mining, and Jabari), plus the royal family. We see this during the prologue and during the first waterfall fight scene.
The War Dogs, however, is the name given to spies that leave Wakanda's borders. This applies to Nakia, N'Jobu, and Zuri. Nakia, for example, is the heir to the River Tribe as well as being a War Dog--we see her decline her chance to claim the throne by fighting T'Challa.
So, a War Dog can be from any tribe or group--it just means that they're a spy.