It's not a generalization; it's an analogue. The map it's analogous to is that on the restricted universal enveloping algebra (thought of as differential operators on the algebraic group) given by pullback by the Frobenius.
Similarly, there's a Frobenius mapping functions on the group to the quantized function algebra which is dual (in the sense of Hopf algebras) to the Frobenius above, just as the Frobenius on the restricted Lie algebra is dual to pullback of functions by the Frobenius under the usual duality of functions on the group and invariant differential operators (the pairing is differentiate and evaluate at the identity).