Before I get into my explaination, I found a couple of interesting historical notes about Apollonaire and this particular edition of Calligrammes: Poems of Peace and War 1913-1916.

On this publication:

Shortly after his death [9 November 1918], Mercure de France published Calligrammes, a collection of his concrete poetry (poetry in which typography and layout adds to the overall effect), and more orthodox, though still modernist poems informed by Apollinaire's experiences in the First World War and in which he often used the technique of automatic writing.

-Guillaume Apollinaire: Wikipedia entry

On Apollonaire's process:

Apollinaire initially sketched his Calligrames in notebooks or on scattered pieces of paper, and then carried his sketches to a typesetter who would set the letters and words on the page following Apollinaire’s design.

- Calligrammes and Expressive Typography

To answer the question, the sample in your original post is most likely manually typeset, possibly with the aid of a Monotype Super-Caster machine. Below are a couple of reasons to back up my claim.

Why not Linotype?

While Linotype was indeed a prominent typesetting technique at the time, and would give you the similar evidence of an industrial press machine at work (it is mechanical letterpress after all), the process, or hot metal typesetting, was developed specifically to produce molds of straight lines or strips of custom type in one block:

The linotype machine operator enters text on a 90-character keyboard. The machine assembles matrices, which are molds for the letter forms, in a line. The assembled line is then cast as a single piece, called a slug, of type metal in a process known as hot metal typesetting.

-Linotype machine: Wikipedia entry

Because of this limitation, we can safely rule out Linotype for any of the Calligrammes in this edition. However, it can't be ruled out for some of the other poems throughout the book that are comprised of straight lines of text.

The Case for Monotype

The Monotype casting machine uses a similar process to Linotype but differs in that it produces individual letters instead of full lines of text:

The Monotype machine worked by casting letters from "hot metal" (molten metal) as pieces of type. Thus spelling mistakes could be corrected by adding or removing individual letters. This was particularly useful for "quality" printing - such as books. In contrast, the Linotype machine formed a complete line of type in one bar.

-Monotype Imaging: Wikipedia entry

While the idea of mechanized typesetting was to produce more precise, straight-ruled lines and blocks of text, the Monotype letters/characters could theoretically be typeset manually if desired:

The Super-caster, another machine produced by Monotype, was similar in function to the Thompson, Bath, pivotal and others casters but designed to produce single type (including even larger sizes) for hand setting.

-Hot metal typesetting : Wikipedia entry

Closing note:

The Calligrammes are an idealisation of free verse poetry and typographical precision in an era when typography is reaching a brilliant end to its career, at the dawn of the new means of reproduction that are the cinema and the phonograph.

Guillaume Apollinaire, in a letter to André Billy (from Calligrammes: WIkipedia entry)

I know this may not be the definitive answer you're looking for, but it might steer you in the right direction if you're not satisfied. This is a beautiful print specimen, thank you for sharing!