User pretty quickly recognize the picker accepts day & month. The ordering is adaptable to locale, and context can be provided to the control as to whether to favor past/closest/future occurrences (eg. bookeeping arrears vs planning future calendar event). Year can always be included if the users wants to be unambiguous.
I'm not familiar with the special 3/9 thing, but if you're serious that the behavior is something you think users in your locale would expect, then it wouldn't be hard to override the handler on a project using the picker and implement that.
This means that as a user, I have to toss a coin. The special 3/9 thing us that it can be either 9 March or 3 September. User locale is only loosely related to how user actually wants to have it or even assume to have it.
I should have clarified, by adaptable to locale I meant you have control via configuration.
The developer can supply a default or even lock it down. E.g. Users of the last app I built incorporating it always expected M/D and the app operated on a kiosk, so that's how we did it.
If no initial configuration is provided, the default does feed from the date format set up in the system locale settings. I agree in practice that's not always guaranteed to reflect what the user wants, but it is the intended spot the OS provides for configuring this and is usually the best source for a default: https://i.imgur.com/KShFPtb.png
If the developer hasn't locked it down, it's easy for a user to override (either via a setting exposed in the app - which you could even attach to a toggle right in the input field's context menu if you'd like, or via a config file / registry setting).
I missed the rot13 thing, and thought "Guveq bs Frcgrzore" referred to some kind of unique cultural event where dotancohen came from (like if I referred to 9/11 in the USA, or typed in "Thanksgiving").
Finally, you could also just type "Mar 9" for a one-off - or use the dropped-down GUI instead of typing anything in the box at all, if keying is not for you.
I'm not familiar with the special 3/9 thing, but if you're serious that the behavior is something you think users in your locale would expect, then it wouldn't be hard to override the handler on a project using the picker and implement that.