Is general covariance a symmetry? If it is, what is its symmetry group and corresponding generator?
It is a symmetry, indeed.
The term, general covariance, could be used in a wide range of context but since the question is tagged in General Relativity, it has a specific usage.
It means the theory is covariant under (infinitesimal) general coordinate transformations $$ x^\mu \rightarrow x^\mu + \epsilon \kappa^\mu (x) $$ where $\kappa^\mu (x)$ are smooth functions of spacetime. This is called diffeomorphism symmetry, which simply means switching a generic reference frame (not necessarily inertial). So, it is the symmetry of relativistic gravitation theories, and locally it reduces to the Poincaré symmetry in General Relativity.
A special case is the translation symmetry, when $\kappa^\mu$ is constant, i.e. does not depend on $x$. Another special case is Lorentz symmetry where $\kappa^\mu$ is proportional to the Lorentz generators. Together they form Poincaré symmetry, which is a sub-symmetry of diffeomorphism, and it is the symmetry of Special Relativity.
Not really a symmetry in the sense of having a symmetry group. The idea is that physical laws must have the same form under different coordinates, or other applicable transformations. So, whatever symmetries the system being studied has, the laws regarding it must have the same form after a transformation. So general co-variance can refer to rotations and coordinate transforms like switching to spherical polar coordinates. Or it can, in relativistic cases, refer to changes of velocity. Or if you are doing electromagnetism, it can refer to gauge transformations.