I just released Exifography 1.1.3, which includes the ability to add unique IDs and classes to the HTML output of your EXIF data.
EXIF block ID
In order to add the image ID to the EXIF container HTML use %s where you want the ID to appear. E.g., <ul class="exif" id="%s">
The output for that ID is wp-image-ID, since WordPress automatically adds the same as a class to the images it inserts into posts you can fairly easily associate the EXIF block with its image, which will be useful for javascript effects and that sort of thing.
If you happen to be using that ID as an ID for something else on the same page you could include it as a class instead: <ul class="exif %s">.
EXIF item classes
You can also add a unique class to each exif item to style them separately.
E.g., <li class="%s">
The class that is applied is the same as the keys used in the shortcode. So, in order to style the timestamp differently to all the other items, for example, you would use .created_timestamp {text-transform:uppercase; color:blue;}.