There is no true definition, since the term is an abbreviation.
This means the definition can vary. However, if you follow the Drop_C_tuning link, you'll find the majority of songs written in "Drop C" are CGCFAD (or CGCF for a four string bass). From a chord perspective, the fingering is the same as Drop D, only at a lower pitch, one whole step down.
In Drop C (or Drop D), the first two strings make 5ths (instead of 4ths) which makes it possible to play chords more like fingering a scale. For example, Outshined, by Soundgarden.
E|----------------------||--------------------------------||
B|----------------------||o------------------------------o||
G|----------------------||--------------------------------||
D|----------------------||--------------------------------||
A|----------------------||o-0-0-3-5-3-0-3-3--6-5-3--(3)/-o||
D|-----------------10\--||--0-0-3-5-3-0-3----6-5-3--(3)/--||
There's been a history of dropping tuning to achieve a deeper sound in blues and hard rock. Robert Johnson and Jimi Hendrix tuned to E flat. Heavy metal and other styles use Drop D tuning. More recent artists drop that down another whole step to Drop C tuning. For example, Chevelle's Face on the Floor,
F|-----------------------------|
C|-----------------------------|
G|-------530----------530------|
C|-0-0530---5-3-0-0530---5-3-0-|
I've also had other musicians that state the following tuning is Drop C (only E string is dropped to C)
Playing with a major 6th between strings would be awkward, even on a bass, but, the terminology could certainly be used for both, since it is a short form.
Having said all that, bass players do not always tune like the guitar players. 4-string bass players may choose to play in a different position, rather than retune, but often can't play an octave down for all the notes. 5-string bass players can be a bit more flexible, using the same tuning when accompanying players using standard or drop tunings and still being able to hit all the notes one octave down. I've even seen one bass player tune in fifths, like a cello (CGDA) for more range on 4 strings.