Where are digital touch messages kept?
The keep option simply means they are not removed from the conversation - just as with audio messages, it's a space-saving measure.

You can long-press on a digital touch item to get a Copy option. From there you can paste it anywhere that accepts photos as paste data (such as the input field in WhatsApp or Messenger, or an image editing app that supports it). Note this will copy it as an image, and not as a GIF/video that contains the motion present in the original.