In what sense is God infinite?

A distinction is made between "potential infinity" and "actual infinity". Potential infinity being the infinity we talk about when we say something is unlimited, or extends without bounds. Actual infinity refers to the size of a set or collection of things, and it is this notion of infinity that Cantor conceptualized with his infinite cardinals.

When we say that God is infinite, I would argue we mean it as a potential infinity, not as an actual infinity. That is to say, God is infinite in that His goodness, mercy, love, what-have-you is unbounded, without limit, or without end. Actual infinity answers the question "how many?". When applied to God, the answer is 1.

If you were to ask the question "what is the size of the collection of concepts that are contained in God's intellect" on the other hand, this would be more amenable to the type of analysis given above. However, in that case, I think the question is incoherent. It is akin to asking for the cardinality of the set of all sets. But there is no set of all sets! And if you want to allow for classes and ask for the cardinality of the class of all sets, the question is not meaningful mathematically.

Furthermore, the mathematical concept of cardinality of sets is reliant on the axiomatization chosen as a foundation of set theory. It is true that the continuum hypothesis is independent of the standard axioms of set theory. However, if one is committed to the notion of objective truth, then one must ultimately decide whether this statement is true or false objectively. Assuming that the notion of cardinality has some form of objective reality, the truth of such a proposition should ultimately be derived metaphysically from that reality. What the undecidability of the continuum hypothesis ultimately shows then is not that there is no objective truth to the statement, but that the standard axioms of set theory are not a complete foundation of the metaphysical reality underlying the concept.

Consider, for example, geometry.  Euclid famously axiomatized geometry with 5 axioms but believed that the 5th axiom (known as the parallel postulate) could be demonstrated from the other axioms.  However, it was later proved that the 5th axiom was independent of the other 4.  This is because while it is true in standard Euclidean geometry, it is false in non-Euclidean geometries such as hyperbolic geometry.  If we wish to use these axioms to describe our universe, it is no longer the case that the parallel postulate does not have an objective truth value (at least if one is a realist, which I am).  Rather, one must determine which underlying geometry obtains in our universe and only then can on say whether the parallel postulate is true or false.

Analogously, the standard (ZFC) axioms of set theory describe some abstract object known as sets.  The continuum hypothesis says something about the relation between two particular cardinal numbers.  This relation is determined by the universe of sets being discussed, just as the parallel postulate is determined by the underlying geometry.   The truth value of the continuum hypothesis, therefore, in relation to the question "what is the size of the collection of concepts that are contained in God's intellect", is to be determined by which, if any, universe of sets obtains in relation to this question.

Of  course, whether or not sets are "real" philosophically is up for debate.  Sets are not simply a universal.  They are an abstraction.  And the metaphysical status of abstractions is anything but settled.

Also note that the continuum hypothesis is about the relationship between two particular cardinal numbers and not about the cardinal numbers themselves.  That this relationship is determined by the underlying universe of sets need not be surprising when considered in this light.  Unexpected, perhaps, but not surprising.