One mostly sees
Kate as a nick for
Katharine;
Kit for
Christopher and therefore a masculine nick. However, it's a small and playful step from
Kate to Kit(ty).
There's a lovely children's book by
Geoffrey Trease, called Cue for Treason, in which Shakespeare is one of the characters! Another is a girl who for excellent reasons runs away from home disguised as a boy and becomes an actor specialising in female roles ... she is a
Katharine who calls herself
Kit while on the run. But how authentic that is, I can't say.