I believe I know quite well OPE and ORE, but I'm unsure about what family to put them in. Can we consider them as a sub family of Functional Encryption, like Attribute Based Encryption or Inner Product Encryption or are they a new paradigm of their own ?
Argument: In such functional encryption, encryption uses a public key. But we can't have secure public-key order-preserving encryption, because knowledge of the public key allows adversaries to encrypt, and ability to encrypt combined with the order-preserving property allows to decipher by dichotomy.
As noted in comment, that argument does not apply to Secret Key Functional Encryption, a notion I'm unfamiliar with. So I reserve my opinion on if OPE can be obtained in the framework of SKFE.