Indirectly, because of Looney Tunes.
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While early steam carriages of the 1800s were quite noisy, there was still enough safety concern that 1865 saw the passage of the Locomotives on Highways Act affirming that "self-propelled vehicles on public roads must be preceded by a man of foot waving a red flag and blowing a horn".
This early precedent meant when motor vehicles proper started to appear, there was a variety of choice in signal, including horns, whistles and bells. In the US, bells were originally the most common, but there arose the problem of them being easy to confuse for other sounds, leading writers to advocate for bulb horns instead; the sound being firmer and from a more obvious source than bells. However, continuous use of bulb horns itself was a problem, as noted in 1913:
The dull, monotonous, and utterly innocuous droning of the bulb horn has become such a continuous noise in many sections and cities that people pay no more attention to it than one does to the buzz of machinery in the building where he may be located.
The closest thing in this time period to a musical horn would be the Gabriel horn by the inventor Claud Foster. He played trombone in a band, which gave him the idea of a replacement for the bulb horn by blowing car exhaust through pipes. He founded his car horn company in 1904; you can see a 1905 model here. However, the horn was "multi-toned" and while playing a pleasing sound, did not exactly "play a tune".
(Foster also invented the shock absorber; his company Gabriel Co. sold 75% of all shock absorbers made in the early 1920s, and his invention made him rich enough to -- in 1952 -- give away nearly $4 million dollars to charity, $45 million in 2021 dollars.)
Despite the wild variety of early car noise experimentation, by the 1930s car horns in the US became relatively standardized into a two-pitch combination, of E flat and C, considered firm but relatively pleasing to the ear. There was the occasional novelty like this 1956 "Kattle Kaller" meant to imitate a bull but more typical, all the way to the 1960s, was something like this horn from a 1965 Mustang (I've linked to a spot where the person in the video tests the high and low notes separately so you can hear them).
However, late-60s -- early-70s is roughly when musical car horn start to appear, common enough that the European Economic Community bans them by 1971. (This trade publication mentions The Colonel Bogey March and La Cucaracha specifically.)
Nearly at the same time was the debut of the Plymouth Roadrunner.
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Robert "Big Bob" Anderson was named VP of sales at Chrysler-Plymouth in 1965, and was looking about for a way to grab the interest of the youth -- the forefront of the muscle-car craze. He asked Brock Yates of Car and Driver for ideas and he mentioned getting a stripped down car with the biggest engine possible (essentially, allowing a huge engine and a nonetheless more economical price); while this had been done before with cars intended for drag racers, there wasn't anything like it for the public.
This idea was appealing not only in opening a new category but in being able to use parts already on Chrysler's shelves. The project got handed of to Planning Manager Jack Smith. Roughly a week into the design of the new car Smith's assistant (Gordon Cherry) told him to watch a certain Saturday morning cartoon; that there was a "perfect name for the car".
Smith did, in fact, watch the actual cartoon -- Road Runner -- and immediately knew the idea would work. The Warner Brother cartoon dating back to the late 1940s showed the hapless Coyote trying to catch a Road Runner but always falling short.
The name was pitched over to the ad group -- who had being going with the name "Lamancha" -- and after being told
Do you ever watch cartoons on television? We think a perfect name for the car is Road Runner.
they agreed enthusiastically.
Smith managed to get the name Road Runner (as a car, it didn't clash with trademark category) but wanted official licensing from Warner Brothers, and managed to negotiate a $50,000 annual fee. This led to an actual Road Runner logo being put on the car.
More spectacularly, Chrysler also wanted the "sound" of the Road Runner -- the "beep-beep" he makes. (The "beep-beep" idea for the original cartoon came from Chuck Jones, when he heard a person outside his office saying "beep-beep" while passing to get bystanders out of the way.) Smith shopped around horn vendors and managed to get a horn with the extra feature only for 47 cents over normal price; the horn is marked in vehicles as the "Voice of the Road Runner". You can listen to the horn here.
The 1968 Road Runner was a smash hit -- far surpassing expectations -- moving 43,000 new cars.
In a strange sense, it was the first "official commercial" vehicle with a custom musical horn, and giving the timing, it essentially invented the popularity of the category. It wasn't much long after you had Fiamm advertise a line of musical choices; in addition to Colonel Bogey and La Cucaracha you could get Never On Sunday, Lili Marlene, and O Sole Mio.
I have not found any oral histories or the like from engineers who explain why they picked the songs they did -- I would guess the ability to the convey the tune without many notes -- but given the likely influence of Road Runner, La Cucaracha seems like a natural pick: it was very strongly associated with Looney Tunes, particularly the character of Speedy Gonazales, the other superfast character from the Looney Tunes group. It also pinged -- in an awkward way -- the Chicano demographic, which was heavily into muscle cars, and customization in particular.
Notice also how the early 1970s ad includes two other "identity picks", O Sole Mio for Italians and Never on Sunday for Greeks. Horn makers later included La Bamba (Hispanic, again) and Dr. Zhivago (Russian). And, as immortalized in The Dukes of Hazzard, Dixie. La Cucaracha reigned supreme due to early market entry and a number of movie appearances, and by The Lost Boys (1987) a weirdly distorted broken version can be used as a joke.
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You may also like my answer to why stereotypical doorbells use a descending third.
Eisenschenk, W. (2017). 1969 Plymouth Road Runner. United States: CarTech.
McLeod, K. (2020). Driving Identities: At the Intersection of Popular Music and Automotive Culture. United Kingdom: Taylor & Francis.