nt.number theory - Equivalence of Finiteness of Class number to Property of Ideals of Algebraic Integers

In the statement of the question you meant "every ideal is finitely generated".

In any event, these two types of finiteness (of ideals as modules and of the ideal class group) are not at all equivalent. For example, you can replace integral closures of $mathbf Z$ in finite extensions of the rationals with integral closures of $F[x]$ in finite extensions of $F(x)$, where $F$ is a field, and thus speak about ideal class groups
in this other setting. There the ideal class groups can be infinite, even though the ideals are still finitely generated. To take a concrete example, consider the integral closure of $mathbf C[x]$ in $mathbf C(x,y)$ where $y^2 = x^3 - x$. That integral closure is a Dedekind domain and its class group is infinite; in fact the ideal class group is isomorphic to the group of complex points on the elliptic curve $y^2 = x^3 - x$, which as a group looks like $mathbf C/mathbf Z[i]$.

Finiteness of class groups is somewhat special (not unique to it, but still special) to the setting of rings of algebraic integers and not valid in general Dedekind domains.
In a general Dedekind domain, any ideal has at most 2 generators and this is unrelated to the size of the class group.