What a good question! And unhelpfully, I don't know Swamp Thing canon other than what I've seen in your in-bar posts and such.

I'm going to say that yes, it sounds different than the rest of Bar's forest. It's a different forest in a different world, and also the Enchanted Forest in canon is... sort of an entity. Not exactly sentient, but nebulously somewhat aware, especially of people it likes (like its king) or threats to itself. And it's definitely bound together, with its own network of magic that's all interconnected and not much connected to anything outside the forest; the boundary of the Forest isn't just where the trees stop. What it actually feels like to your pup in any detail, though, I'll leave up to you.

Does that help?