The translations in the CFT are a part of the conformal group, so they're obviously identified with an isometry (diffeomorphism that preserves the metric everywhere) of the bulk!
For the $AdS_{d+1}/CFT_d$ correspondence, the conformal group is $SO(d,2)$ and may be obtained by extending the Lorentz group $SO(d-1,1)$ by the $d$ translations, $1$ scaling generator, and $d$ generators of special conformal transformations. The AdS spacetime has this isometry because it is a $(d+1)$-dimensional hyperboloid of a sort so its isometry is a noncompact version of the $SO(d+2)$ isometry of the sphere $S^{d+1}$.
To see that $SO(d,2)$ contains $d$ mutually commuting translation generators, find a null direction in the $(d+2)$-dimensional space, $x^+=(x^0+x^{d+1})/\sqrt{2}$. Then the generators $J^{i+}$ for $i=1,2,3,\dots d$ commute with each other because $g^{++}=0$.
When an excitation or defect is inserted to the boundary CFT, the state is no longer invariant under some/all translations. The AdS statement equivalent to it is, of course, that the state of the bulk is no longer invariant under the corresponding isometry.
It isn't really an objectively well-defined question to ask "what is the unbroken residual bulk diff invariance". The diff invariance is a gauge symmetry which is an auxiliary tool. We start with a redundant description (redundant space of configurations classically; a redundant, larger than physical Hilbert space quantum mechanically) and we impose identifications given by the gauge symmetry to reduce it to the physical space (configuration/phase space classically, physical Hilbert space quantum mechanically).
There is no "the" right set of gauge symmetries we have to use for a given physical system. That's made clear by the AdS/CFT correspondence, too. The two sides employ very different gauge symmetry principles to reach the final product. The CFT uses e.g. the $SU(N)$ local/gauge Yang-Mills symmetry; there's no trace of it in the bulk description. On the contrary, the bulk theory employes higher-dimensional diffeomorphisms etc. to eliminate the unphysical graviton polarizations, aside from similar gauge symmetries for other bulk fields; there is no trace of diffeomorphisms in the Yang-Mills description.
But whenever you use the covariant gravitational description, the group of diffeomorphisms is completely given by all the coordinate redefinition and this group is completely determined by some general data such as the spacetime topology. We demand the physical states to be invariant under (i.e. annihilated by the generators of) the diffeomorphisms that converge to the trivial map at infinity (near the boundary). The isometries of the AdS space i.e. the conformal transformations are not converging to the identity in the asymptotic region which is why they don't have to keep the physical states invariant.
