Most of these enclosures contain a standard SATA drive. It is mostly the later ones where they solder in a proprietary USB controller.
The only way to find out is to shuck the drive with a spudger tool (either plastic or metal spatula). A guitar pick usually does the job. Check under any labels for hidden screws first. Expect to completely destroy the enclosure in the dismantling process. But you've nothing to lose here, and it won't affect the data on the drive.
Once you access the bare drive, you can inspect the interface to determine what to do next. If it's a small USB controller PCB plugged into a standard SATA port, then just pop the drive in a new enclosure or dock. If it's a soldered USB controller, then it's time to send it to professional data recovery.
Alternately you can just skip all this and pay professional data recovery to, um, recover the data.