At least for English language material, it is principally a product of interwar romanticism about the Legion, which is far and away best exemplified by the book Beau Geste, published in 1924 by the English author P. C. Wren, and which tells the story of three English brothers who are gentlemen, yet in hard times, under suspicion for a crime they of course didn't commit, and one... yes... unable to pursue the romance he wants. So they go off and join the Legion to start a new life. Several films in the period were also made, the big one being a 1939 flick starring Gary Cooper. Whether Wren himself ever served is a matter of debate with no clear answers as identities in the Legion were of course masked.

The image that this and other works pushed is well summarized by Douglas Porch in the introduction to his work on the unit as:

[A] mongrel unit recruited among the scrapings of humanity, perhaps even among the criminal classes, but also containing a fairly sizable contingent of “gentlemen rankers” who fled into the Legion to escape a dark past or out of a desire for adventure.

That isn't to say there weren't men there who were really driven by those reasons... accused of a crime, seeking a thrill, escaping a jilted romance... This wasn't something created out of whole cloth, as it was a reputation that the Legion had been cultivating in the years before WWI (although more common in French speaking circles than English), and buoyed by its very real policies about identity with the famed anonymat, but they were a very small minority. Far more common were men who had nothing else in life, and looked to the Legion as an option of last resort - "victims that misery or hopelessness have thrown into the chasm of the Legion" as one writer described most of their number.

But it is the "Beau Geste Romantics", as Porch calls them, that fixate the imagination. The "men of good family and education who after being jilted or enduring some other downturn in their affairs sought to bury their misfortunes, and themselves, in the Legion". It makes for a compelling story, as seen with Beau Geste, not to mention an ensuring one, as seen with the parodies such as Looney Tunes that you mention, or Laurel & Hardy's Beau Hunks.

I think it is important to keep in mind that "wanting to start a new life" is similar to the "joining due to heartache". Both cases are men trying to get away from something in their past that disappointed them, and find a new start, which the Legion appeared to offer. Another writer of the period, who had served in the early 1900s, broke the Legion into four broad groups of men:

1. The miserable, without work who prefer to enlist rather than beg and sleep under bridges.
2. Those who seek adventure, generally the very young.
3. Deserters.
4. Adventurers who come to forget their mistakes or who would like to be forgotten by society. Romantic hews spring up sometimes, and it is these cases, purely exceptional, which give rise to the legend that kings having lost their thrones, bishops who misplaced their miters or generals who lost their stars make up the majority of legionnaires.

But while that last group might have been tiny, it is important to keep in mind that the Legion itself liked to cultivate that image, as it of course helped with their reputation. As Georges Manue put it, “The legionnaires are mythomaniacs, inventors of fables which they are the first to believe in”. After all, they might mostly be petty criminals on the run or the poor driven to no alternative, but it is much nicer to have other people think maybe you are the younger son of an English Lord mending your broken heart.

The irony too is that Beau Geste was not popular in France. The 1926 and 1939 films were both prohibited in France for quite some time. Although the story in many ways painted that oddly romantic image of the Legion, it also had the heroes desert, after all, and we can't be having that, now can we! The French authorities would have none of that. Even though much of the romanticism leaned into tropes already present in French and Legion discourse, it was a bridge too far, but it had little impact, and it is likely that in a bit of the tail wagging the dog, the portrayal of the Legion in that way helped boost recruitment of English speakers in the late '20s and early '30s as men took that image of the Legion to be closer to reality than it really was. More than a few were in for a rude awakening.

So anyways, the sum of it is that the image of the jilted lover running off to drown his sorrows in the Legion is part of the larger trope which, in English language literature, we see principally rising up in the 1920s from Beau Geste, which sought to paint the Legion as a haven not only for the scum of the earth, but a certain kind of English gentleman, a man of true honor, and, to return to Porch's eloquent description, "bolstered the notion that behind every legionnaire lay a story of more than ordinary interest, suggested an upper-class presence in the ranks and delivered on the promise of redemption that Legion service was meant to bestow."

Sources

Porch, Douglas. French Foreign Legion. Skyhorse Publishing, 2010.

Windrow, Martin. Our Friends Beneath the Sands: The Foreign Legion in France's Colonial Conquests 1870-1935. Orion, 2010.