My car belongs to me. We don’t use the possessive pronoun ‘my’ in the third person or when speaking generally. ‘No my car day’ sounds like nonsense to native English speakers. To express this we would probably say something like, ‘No Car Day,’ just as there is an international No Smoking Day. Perhaps more accurately we would say ‘Public Transport Day.’ No my car day is Japanese.

ノーマイカーデー

「My car」は私の車という意味です。「My」は他の人の物には使いません。「No my car day」はネイティブスピーカーには意味が通じません。通じるなら、「No Car Day」です。国際禁煙の日「No Smoking Day」と同じ風に言えばいいです。「公共交通の日」と呼んだほうがわかりやすいかもしれません。「ノーマイカーデー」は日本語の単語です