Suppose we define a CFG such that it is possible to produce strings of the form $a^nb^n$ (in this case, I would think, we would need epsilon-productions). Then one such $L(CFG)$ is $a^nb^n$.
However, let's say it is also possible to produce strings not of the form $a^nb^n$ from this grammar.
Can we still call $a^nb^n$ a language generated by the grammar, even if it is not the only/unique language accepted by the grammar?
In other words, is the language accepted by a CFG the set of all strings that can be derived from the grammar? Or can it also be a proper subset of these strings, fulfilling some desired criterion?