Absolutely not! Do not, for any reason, format that drive. This is a false message that Windows gives when something is wrong with the USB interface for a mass storage device.
I is difficult from here to know exactly what happened. But this is a problem that other people have reported many times over and over.
There just is something about the USB system and external drives that just does not seem to work right.
Is this on a laptop? If so, go find a friend that has a desktop PC and figure out some way to hook the external drive into the desktop. But do not use the USB adapter. Instead, find some way to hook that drive directly into the desktop. Most likely the drive is perfectly all right and all your data is still there. By the way, is this thing still under warranty? If so, you need to contact the manufacturer of the enclosure. Was this an all-in-one enclosure? I mean to say was the device and the drive purchased as a single unit from a vendor? Is it as Seagate product? I mean is the entire package got a Seagate brand on it? Seagate is very good about standing behind their products. But that may not be the case for enclosures that are made and packaged by others.