I agree with Point 1 - each individual has their own tonal 'fingerprint' (literally), from their physiology to their style of play - so it is all in the fingers (or pickstrum thingy). Another factor is that we all have good and bad days with the bass - some days I'll sound amazing (to my ear) and others I can't do a thing right! So maybe its also in the mind?
Regarding Point 2, I suspect that the audience don't care what the bass sounds like as long as its not too loud or harsh sounding, so the gear / backline / monitoring / PA is immaterial to them. We choose our own gear based on what we like regarding the sound, look or reputation; having equipment that we trust or have confidence in could help us with the 'in the mind' part of Point 1 - if its sounds good to the player, its one less thing to fret about at gigs, allowing more time for enjoying the moment.
Obviously though, if the equipment just doesn't function properly, its going to be stinky-poo!
*plus I like shiny things!