No, it is not possible to get some constant value/bytes from U2F device that will be unique for a particular device and can be used later as encryption key, mainly because of FIDO Privacy concern: "identifying devices would reveal a unique identifier for a device across unrelated origins, violating the user’s privacy." (c) FIDO specs.
In another hand, this is why U2F Signature feature also cant be used to get some constant value that can be later used as an encryption key:
- U2F device sign a Digest that is created by Client and Device (U2F adds signature counter to Digest). So every new signature will be unique even if Client provides the same data twice.
- When U2F device generates a new KeyHandle (ECDSA pair) it uses its own random value. So it will never generate the same KeyHandle even if you provide the same data for the 2nd request.
By FIDO philosophy, the U2F key mainly designed to ensure a Trust between Authentication Service and remote client. When you want to adopt U2F key for other authentication/security scenarios take care about a dozen of attacks that can be applied.