Patty, not Paddy. I've definitely heard it called Paddy's Day (we use that here ourselves and we're often referred to as Paddy's ourselves lol) but never saw Patty before.
Seeing Patty makes me think it's trying to feminise it which I don't really get. If it's short for Patrick, I don't really get it either tbh.
Regarding the verbalising of Paddy and Patty above, I think depending on your accent, they aren't necessarily alike.
I've never seen or heard of any Patrick having Patty as a nickname tbh. Paddy comes from two areas that I know; Paddy being the nickname for Patrick and more relevantly Pádraig.
It's also the slang term for an Irishman but originated from the British as a derogatory term for us, most likely taking exception with our language (use of Pádraig)