kitty [coll.] (kitten) - das Kätzchen
kitty [coll.] - die Mieze [ugs.][fam.]
kitty cat - das Miezekätzchen
kitty litter [coll.] - die Katzenstreu
kitty flu [med.] - der Katzenschnupfen [Tiermedizin]
puss - das Miezchen
puss - die Mieze
pussy - das Kätzchen
pussy - die Mieze
pussycat - die Miezekatze [fam.]
pussy (Amer.) [sl.] [pej.] (effeminate male) - der Schlappschwanz [ugs.] [pej.]
pussy (Amer.) [sl.] [pej.] (effeminate male) - der Waschlappen [ugs.] [fig.][pej.]
As far as I am aware, pussy (cat) is still a/the normal expression for cat among younger British children
The nearest meaning I can think of is something like "kind", "gentle", "considerate" often used of a man who is normally thought of as intimidating, as in: "He's a real pussy when you get to know him."
LEO übersetzt pussy unter anderem mit
der Schlappschwanz [ugs.] [pej.]
der Waschlappen [ugs.] [fig.][pej.]
I wouldn't say that that meaning is very BE, btw.
re 'pussy' meaning a cat (cf. esp. 51, 57)
Yes, there is a BE/AE difference. In Britain we hardly ever use 'kitty' in this sense.
When I was in America recently someone mentioned that she had to 'pick up the kitty' (from the vet's) ...related discussion: Pussy Riot - Muschi-Krawall