If you cover reputation change in the achivements dialog by mouse pointer - tooltip with timepoint will be shown:
For badge items no such behaviour exists:
If you want to know that date you need to find this badge in the user profile page, click on it and find one (because you can have many) with more suitable date (according to dates for nearest rep changes in the achivements dialog). It's a little bit tedious.
My request is to add tooltip with timepoint for earned badge items in the achivements dialog.