In the Cooper Pair box, the conjugate momentum $\hat{n}$ of the reduced superconducting phase variable $\hat{\varphi}$ takes only discrete, integer values due to the $2\pi$ periodicity of the phase. As this conjugate momentum is the relative charge number on the island, this corresponds to the discrete charge on the island.
In the inductively shunted CPB, the inductance destroys the periodicity of the potential in $\varphi$, we have a $\frac{1}{2}E_L \varphi^2$ term additional to the usual $\cos{(\varphi)}$ term of the Josephson Junction.
Now Girvin (http://www.capri-school.eu/lectureres/master_cqed_les_houches.pdf, after 4.41) or Koch et al. (PRL 103, 217004 (2009)) write that this leads to the charge variable becoming continuous as opposed to integer-valued like in the CPB.
Formally, this makes sense to me: As we do not have periodic boundary conditions for $\varphi$ anymore, we will not have discreteness of the conjugate momentum, i.e. the charge anymore.
I wonder, however, how to interpret this physically: After all, the transferred charge must still be quantized in the form of Cooper pairs. So what is meant by the continuity of the charge here?
