My school jazz band's bass player has a 5 string bass but it is tuned like a guitar with a broken 'E' string, but an octave lower. This means that the highest string is a 'B'. Is this common? I know that most 5 string basses have an extra lower 'B' string - like what a 7 string guitar would have. Is this some modification that people usually have?
Not that common. Normally 5-string basses will have a low B. Giving B E A D G.
The other option is to have a high C. Working on the premise that every string is a P4 from the next. Giving E A D G C.
It is probably the result of a guitarist being more used to guitar tuning. But it doesn't help bass players. The idea behind the G>B (M3) on guitar is to facilitate chord playing, which isn't usually what a bass player is expected to be doing. Yes, I know we do, but it's not generally in the remit!
I'd be inclined to either tune it E A D G C, or change strings to B E A D G, which is a much more useful arrangement. Not necessarily to play very low notes, although that's a bonus (especially for tunes in D or E♭), but to have a 5th of a chord underneath, on that B string, to save moving down the fingerboard.
You're right about 6-string basses - they usually go B E A D G C. I used to play with a guy who used an 8 string - F♯ B E A D G C F...