As usual in Statistical Mechanics, different ensembles provide different values for thermodynamic and statistical properties. It is only for a subset of the possible Hamiltonians and suitably large systems that these different values go to the same limit. The technical procedure is the so-called thermodynamical limit.
Once one knows that the thermodynamic limit exists for the Hamiltonian of interest, it is usually possible to show that different ensembles provide the same thermodynamic and statistical descriptions independently on the starting ensemble at the thermodynamic limit. This the case for the ideal gases (bosons or fermions).
We also note that even though the thermodynamic limit implies an infinite size, numerical deviations from the limit values can be very small, even for a few tens of particles if periodic boundary conditions are used.
The above considerations apply as well to the ideal gas both in the classical and quantum regimes. In the case of the ideal gas, it is possible to show explicitly how the difference of description in different ensembles vanishes at the thermodynamic limit.
Thus, in the usual introduction to the behavior of Fermi and Bose gases, the equivalence of the ensembles is taken for granted, and the most convenient ensemble for calculations is chosen. In the case of quantum statistics, the Grand-canonical ensemble gets rid of the constraint about the number of particles in the summations simplifying the mathematical manipulations. However, it is possible to do calculations in the canonical ensemble and you may find indications and links in the answers to this related but different question.
Moreover, a derivation of the Bose-Einstein distribution also in the microcanonical ensemble can be found here.
Summarizing, the answer to your question is that in the case of systems made by a few particles we do expect deviations from the grand-canonical expressions even for the average occupancy of a state. Those deviations can be expressed in terms of a finite-size correction to the chemical potential.