In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. Denoting this composition by $Ax^2 + Bxy + Cy^2$ (so skipping the extra "2" in Gauss's way of writing forms), Gauss notes that $A$ is determined by his definition, while $B$ is determined modulo $2A$. Once $A$ and $B$ are determined, $C$ is determined because the determinant is fixed.
This can be rephrased by saying that Gauss composition is well defined on the equivalence classes of forms under the action of the subgroup of $\mathrm{SL}_2 (\mathbb{Z})$ consisting of matrices of the form $\begin{bmatrix} 1 & m\\0 & 1\end{bmatrix}$. These classes then form a countably generated infinite Abelian group.
This group was studied for positive discriminant by Lenstra in his 1980 paper "On the calculation of regulators and class numbers of quadratic fields". Among other things, he embeds the group in a topological group which he notes is a subquotient of the idèle group of the corresponding quadratic field. Schoof later pointed out in "Computing Arakelov class groups" that Lenstra's topological group is essentially the Arakelov class group of the field.
My question is, was this group studied in its own right after Gauss? Or did all number theorists of the 19th and early 20th centuries study only the class group, and later the group of fractional ideals?