These games all have aim assist, at least on consoles anyways.
The feature that you’re talking about that battleborn has and the other do not is what I would call “sticky aim” (probably already a name for it somewhere but that’s what I just came up with ¯_(ツ)_/¯) In battleborn when you ADS in range of something your aim is heavily affected by the gravity of aim assist pulling you to the center mass of the target you are aiming at. Although the other games mentioned technically have this it is only enabled in PvE modes and not PvP for the most part.
Honestly it might need to be removed. For example when I play Marquis I’m stubborn and if I’m sitting in the ledge on Overgrowth I don’t care what’s coming at me I don’t feel like moving from my spot. Sometimes I’ll be rushed by a Melka and I even manage to kill her at close to point blank range (which is Melka’s strong suit and Marquis’ weak point) However, by repeatedly using the LT (ADS) I can land my shots most of the time.