The date picker box doesn't respect general and look and feel configurations (only in the old issue view)
- Go to General configurations > Advanced settings > Change the jira.date.picker.java.format and jira.date.time.picker.java.format to MMM/d/yy
- Go to Look and Feel > Date/time formats Change the Day/Month/Year Format
- Go to an issue and use the date picker and select some date
The date will be set as MMM/d/yy
Type the date manually or use the new view.