Kingdom of Argonia? I'm assuming they are speaking metaphorically, as I'm not aware of any literal state established within the Black Marsh. Just a tribal system, I had thought.
Anyway, I would think that the various tribes would accept it to varying degrees, but the limited Imperial presence in the western fringes would stop it where they could. Though it seems like a difficult practice to stop if both parties (parents and Brotherhood) are being secretive about it. So, the locals see it as "legal", but the small number of semi-Imperial areas would have it outlawed. Even though it wouldn't make much of a difference, seeing as how succesful the Watch is at catching the Brotherhood in Cyrodiil
The locals may see the shadow scale giving as legal, but I still don't see it being a Morag Tong-Morrowind situation. The tribes are still societies in seems. Really, the Dark Brotherhood isn't accepted by any law. I doubt the tribes would all openly accept the brotherhood.
The brotherhood probably doesn't have a strong presence in Black Marsh anyhow. Place is poisonous and deep. As far as shadow-scales sold into the brotherhood by birth, they are "deported" and raised in other areas. The Brotherhood is really all about wanting people dead. No one really wants the Argonians dead in my opinion. They don't fight many wars, except for with the Dunmeri, and the Dunmeri want to enslave them, not kill them. Even if the Brotherhood did assassinate some in the Black Marsh, I don't see the number being higher than the other provinces (probably lower, even), and I definitely don't see them being legal.
Any straight society, tribal or noble (or something else), wouldn't really accept a death cult as legal in my opinion.