Is it possible to redownload enterprise apps through Apple ID's? To me, it seems as if this should be possible. With a normal app, you can go to 'Purchased' and re-download any previously downloaded apps. 

Is it possible to do the same with an enterprise app which was not downloaded through the app store?
No, if it was not originally purchased/downloaded from the Apple App Store, you will not find it in the Purchased section.

Usually, enterprise apps are distributed from that company's 'app store' only.  If they are deleted from your iOS device, you can revisit the company app store and re-download the app again.  It's up to that company if they want to have a Purchased tab or some other way of seeing previously downloaded apps.