Pickups don’t all sound the same. But the differences are usually subtle. I don’t personally know many people who could tell the difference between a P and a J, let alone 2 structurally similar pickups. The lesson is if it sounds good to you it is good.
My thought on the on the OP was about pickups being different. And while my experience is yes, there can be Ab noticeable difference between them, the other electric components, IE capacitors, tone pots, and especially onboard preamps have a much more profound impact on the tone of a bass.