Ah ancient toilets, I have stumbled on them several times the last year while researching the urban Roman water distribution.
Anyway the sponge is all but unknown to us, it was never a favourite subject in ancient times by some reason. However we believe that it is likely that they were common at public latrines and probably publicly shared in such facilities. Almost all public latrines had a little channel in front of the seats that is assumed to be where these sponges (on sticks) were "cleaned".
On sources; I've actually only found two sources on the use (Seneca Ep. II 70.27 & (Martialis XII 48.7). That you have found a reference to such a sponge in John (19:21) makes it three and I tell you, some major sharks in the ancient water would probably love to get to know this (as A. Wilson).
Seneca:
"For example, there was lately in a training-school for wild-beast gladiators a German, who was making ready for the morning exhibition; he withdrew in order to relieve himself, - the only thing which he was allowed to do in secret and without the presence of a guard. While so engaged, he seized the stick of wood, tipped with a sponge, which was devoted to the vilest uses, and stuffed it, just as it was, down his throat; thus he blocked up his windpipe, and choked" (Loeb)
Martialis:
"Yet your dinner is a handsome one, I admit, most handsome, but to-morrow nothing of it will remain; nay, this very day, in fact this very moment, there is nothing of it but what a common sponge at the end of a mop-stick, or a famished dog, or any street convenience can take away." (Bohn's Classical Library (1897))