So, I adopted two kids, Hroar and Lucia. While Hroar was still an only child, he brought home a mudcrab and I let him keep it as a pet.

Lucia however has never brought anything home. So, I brought home a dog to see if she'd keep it. What happens instead is that Hroar asks if he can keep it, even though he already has a pet.

Is this permanently like this? I keep telling him "no, not right now," but he keeps on asking and Lucia doesn't.

If I tell him "no pets ever" will that disable both of them asking?

As of now, I assume it's glitched.

EDIT: Also I am on Xbox, if that matters.

What you could do is make a save somewhere well away from your residence (i.e. Riften), fast travel to your place and then see what happens. If Lucia does not initiate the pet dialogue, re-load and do again.

*Be warned it might take a long time, but eventually she should ask about keeping the dog.

That's exactly what I'm doing right now. Hroar has been the only one asking, and I'm on try #10.

So this time I told him "no pets ever" and I'm going to try again.

Iirc, saying "no pets ever" will not allow him and, more importantly, her from asking about the dog again.

Like I said, it is a pain in the rear to get the other child to ask about the dog, but keep trying.
*Again, make sure your save is well away from your residence... if you saved after arriving at your place Hroar will keep asking about the dog. Best thing to do is FT to your home, then do a quick FT to Riften, save, and then FT back to your home as that would reset the child encounter/dialogue.

Well, I did exactly like you said, and it actually worked. Thanks a million!