Not exactly. In this case what will happen is you’ll have exactly the same signal doubled. That’s fine and if identical processing is done on both it should sound normal. However, when you add another plugin to one track it’s going to add a small amount of latency (very small, measured in samples, but it will be there). This will delay the phase of the second signal a small amount, and if it becomes significant depending on how much you may run in to phasing. There’s a few ways to deal with this if it happens.
Also with a distortion effect it may actually sound cool if it happens.
The other effect you mentioned is more of a natural chorus/phase effect you are adding because you have two basses playing the same notes but one will be naturally detuned from the other a tiny amount and additionally the timing will be a little different.