For an authoritative answer, I contacted the professor of history at the University of Toulouse, Jacques Krynen, who has written a biography of Guillaume Maran. His response [*] to the puzzling inscription on Maran's statue in de Salles des Illustres strengthens the case that it was an error. My (unsubstantiated) guess is that this happened when the 17th century statues were reconstructed in 1892 after having been destroyed by a fire a few years earlier (as described here).
[*] Il y a bien longtemps que je ne me suis pas rendu salle des illustres... en tout état de cause Guillaume Maran était professeur de droit. Pas mathématicien! (I have not been to the hall of celebrities for quite some time... in any case, Guillaume Maran was professor of law. No mathematician!)
As explained in the comments, I disagree with this analogy. Nonetheless, there is a way you can realize the idele class group (not the ideles) as a group of line-bundle-like objects under the tensor product.
First, some context.
Suppose we have a number field $K$ with ring of integers $O$.
We have a surjective map from the idele class group of $K$ to the Arakelov class group, and another surjective map from the Arakelov class group to the ordinary class group.
The ordinary class group is isomorphic to the group of line bundles under the tensor product. The Arakelov class group is isomorphic to the group of Arakelov line bundles under the tensor product (these are described in Neukirch’s Algebraic Number Theory but he calls them "invertible metrized $O$-modules"). An Arakelov line bundle is basically a locally principal $O$-module $M$ with an inner product (aka metric) on $M \otimes_\mathbb{Z} \mathbb{C} $, and forgetting this inner product corresponds to the surjective map from the Arakelov to the ordinary class group.
To associate a sort of line bundle to idele class group elements, we need to add a bit more structure to our Arakelov line bundles. Then, like before, the surjective map from the idele class group to the Arakelov class group corresponds to forgetting this extra structure.
In order to describe it, I first need to give a different perspective on what an Arakelov line bundle is.
Let $A$ be a one-dimensional $K$ vector space. For every place $\nu$ of $K$, choose a norm on the completion $A_\nu = A \otimes_K K_\nu$ which agrees with the corresponding valuation at $\nu$, so that any nonzero element of $A$ has norm $1$ at cofinitely many places, and the norm on $A_\nu$ has the same image as the valuation on $K_\nu$. The resulting structure, of $A$ plus a choice of norm for each place, is equivalent to an Arakelov line bundle (the elements of $A$ with norm at most 1 at every finite place are the $O$-module, and the norms at the infinite places tell you the metric).
We can specify a norm on a completion $A_\nu$ by choosing a nonzero element of $A_\nu$ and requiring it to have norm $1$. This uniquely determines the norm on all other elements of $A_\nu$. So to get an Arakelov bundle, it’s sort of like we picked an element of $A_\nu$ for every place $\nu$, and then forgot everything except for the induced norms.
To get the extra structure I was referring to, we simply refrain from forgetting this extra data. Let’s define an “idelic line bundle” to be a one-dimensional vector space $A$ over $K$, plus a choice of nonzero element $a_\nu$ of $A_\nu$ for each place $\nu$, so that for any nonzero element $a$ of $A$, $|a / a_\nu|_\nu = 1$ at cofinitely many places $\nu$ (note that dividing vectors is ok here since the vector spaces are one-dimensional).
There is an obvious notion of tensor product and of isomorphism of these line bundles. It is straightforward to show that the isomorphism classes form a group under the tensor product, and it is isomorphic to the idele class group.
I’m pretty sure we can also get a sort of “Picard group” isomorphic to a ray class group in a similar way but I will leave that as an exercise.
Also, this construction works just as well for global fields of positive characteristic.
Best Answer
I know nothing about work of ``idelic nature'' by Von Neumann or Pruefer. Already in the 1930's Weil understood that Chevalley was wrong to ignore the connected component, because Weil understood already then that Hecke's characters were the characters of the idele class group for the right topology on that. I don't know of any place before his paper dedicated to Takagi where he defined the ideles explicitly as a topological group, but he must have understood the situation way before that
When I wrote my thesis I used what seemed to me to be the obvious topology without going into the history of the matter.