Hello,
In the specific screenshot you posted it’s in the dashboard designer itself which cannot be limited beyond what the time dimension allows. However, what is added to the dashboard can be limited. For example, you could add a single calendar filter:

Then you could customize the properties to only allow selection at the day level. You can also further restrict it to hide un-selectable levels, but in order to do that you must set a parent member since we can’t display all of them:
The result of this would be this control on the dashboard:
As you can see the control only shows January 2008, and only allows selection of the day level.
If you really want to customize it for the dashboard developer, as mentioned above you can do it by customizing the time dimension itself. In that case you would uncheck all levels except for the day level. It should be noted if you do that and use the time dimension with data, it would only aggregate to the day level as well.