Yeah I played them both in beta, own them both now, and they don't play anything alike. Overwatch plays exactly like TF2 or Paladins and Battleborn plays like Smite in first person. They share unique characters and are team based games but past that they have very little in common. You'd literally have to change the entire game to add a Battleborn mode to Overwatch. Everything from leveling up, the time to kill, respawn timers, currency accumulation, recruiting NPCs, building structures, creeps, lane based play, etc. in Battleborn make them extremely different. And it's not just reviewers who are reluctant to call them similar it's the majority of people who've played them as well. Randy Pitchford doesn't call Battleborn a moba because "they are F2P and top down RTS" according to him, and it has various gameplay modes which makes it hard to describe, but the games main draw is more a MOBA than HotS.
And since when did team based objectives in a first person shooter require a MOBA mindset? Again, TF2 isn't a moba, was doing this long before MOBA craze and Overwatch borrows heavily from it gameplay wise. There is a reason why people call Overwatch things like "Blizzard's TF2", "TF2 2.0" or "TF2 with waifu" and not "Blizzards DotA 2.0".
I didn't get codes for any of the bonus stuff except the Widowmaker skin. The rest of the stuff, like Tracer, Mercy Wings, etc. all just showed up in the other games when I logged in. In Diablo 3 you have to log into a character before you get the wings.