Q: Is there any reason how and why the bass got tuned in fifths (B, E, A, D, G, C) as opposed to the guitar’s tuning (E, A, D, G, B, E)?
A: Actually, those are fourths the bass is tuned in.
If my memory serves, the bass tuning came first. Although, hundreds of years ago, it WAS tuned in 5ths!
All basses were tuned similarly to violins, cellos, etc. The bass is just the bigger version of those instruments and all were tuned in 5ths (as the rest of them still today). In fact, if you go back to learn some early classical works for double-bass, it is much easier to play when tuned in 5ths, as this was the intended tuning.
I’m not sure at what point people began to tune the bass in 4ths, but it stuck.
The guitar is tuned the way it is (in 4ths, but with that one major 3rd) in order to facilitate chord fingerings. Try tuning it in straight fourths and play an open G chord… ain’t quite the same, is it?!
This concept got taken further with the advent of “open” tunings, which would allow chords to be barred with just one finger.
You can tune a bass (or any instrument) however you like, though! I’ve often had fun by tuning my 6-string bass with that major third (like a standard guitar tuning) and played guitar tunes on it. Michael Manring has taken this concept further than anyone I know of. Alternate tunings on bass are becoming quite common.
Many Gospel musicians tune their basses down (still in fourths, just tuned a step (or more) down. Solo bassists are especially fond of alternate tunings.