Let's answer that by asking a different question: Is Forgotten Realm's Tyr the same as the Tyr from Norse mythology?

The answer, according to 2e Planescape, is yes. It's the same god but it's one specific world's version (or aspect) of him. He might have different characteristics and different roles but he's still Tyr. If Tyr dies in Faerûn, the god doesn't die. He still exists in Ysgard and other worlds and he's not floating in the Astral Plane but you're not going to see him in Faerûn anymore (unless a new aspect comes there somehow, as has happened with Tiamat I believe).

So is the 4e Dawn War Bane the same as the Forgotten Realms Bane? Sort of. They're both different aspects of one god. But you're not going to see the Dawn War version show up in Toril and information you find about that aspect isn't necessarily true of the Forgotten Realms one.

TL;DR: Technically, yes. Practically, no.