Maybe you can adopt another point of view: the class group measures the failure of ideals being principal.
This is more meaningful:
- if the class group is $\Bbb Z/2\times\Bbb Z/2$, then there are ideals that are not principal, but the square of any ideal is principal.
- if the class group of $\Bbb Z/4$, then it is even possible that the square of an ideal is still not principal.
Note however that the class group does not measure the number of generators needed for an ideal, as any ideal (of an integer ring) can be generated by two elements.
To expand more on the unique factorization part:
Let us recall that the name "ideal" comes from "ideal numbers", which were invented by Kummer when he noticed that unique factorization does not hold for the integer ring of a cyclotomic field (which was used in a famous "proof" of Fermat's last theorem).
Kummer then found that, if we add some more "numbers" to the integer ring, then we can have unique factorization. Today we understand that Kummer was talking about the unique factorization of ideals of a Dedekind domain.
These extra "numbers" are what he called "ideal numbers". Note that usual numbers (i.e. elements of the integer ring) can be viewed as ideal numbers by identifying a number with the principal ideal it generates.
It then makes sense to say that the class group measures failure of unique factorization: the class group measures "how many" extra ideal numbers should be added, so as to achieve unique factorization.
E.g. if the class group has order $4$, then we should add three times more ideal numbers to the usual numbers.
Moreover, the group structure of the class group reflects "how" the ideal numbers should be added into usual numbers. This is a bit harder to visualize, but imagine something like $\Lambda = \Bbb Z(1, 0)\oplus \Bbb Z(0, 1)$, and two ways of "adding three times more numbers": $\Lambda_1 = \Bbb Z(\frac 1 4, 0) \oplus \Bbb Z(0, 1)$ vs $\Lambda_2 = \Bbb Z(\frac 1 2, 0) \oplus \Bbb Z(0, \frac 1 2)$.