I can confirm what appears to be "odd" behavior in Microsoft Project. The problem is that the current date display is set to the W4/4 formatting. The purpose of this date format is to hide the real dates in all date-related columns, such as the Start and Finish dates. When you click on a Start or Finish date for any task, even with the W4/4 formatting applied, Microsoft Project will display the real date that is hidden in the background. When you click on any item other than a date, the real date will disappear and will be displayed again in the correct W4/4 date format.
There is no way to "fix" this problem because it is the default behavior of Microsoft Project with this unusual date format. BTW, I ran into this issue myself when I was shooting the following YouTube video:
https://youtu.be/59D1VcP5lrQ