Perhaps but in all likelihood, perhaps not. Who knows? At this point, it is unclear, but people love to jump on the bandwagon and claim ridiculous things no matter how silly. Popular mythos is commonly mistaken for proof.
But it seems clear that "My Dog Has Fleas" isn't from a song, it is just a bunch of words, that alone tell you nothing about the tuning of a uke, unless you have previously memorized the tones associated with the phrase. You might as well say - Gifts Come Every Afternoon. Then at least you could take the first letter of each word and learn the notes. But it's not as silly as My Dog Has Fleas, so it will never catch on.
It's really not as random as you think
It's pretty well researched, and well known throughout the Hawaiian Islands. You're taking to Native Hawaiians whose families have played Uke for generations and handed down techniques and stories. Everything Hawaiian is handed down Orally. Making things up isn't Hawaiian, neither is jumping on bandwagons or lying about things. There would be no Hawaiian culture were that the case.
And not just from Hawaiians but from Locals, who were immigrating at the time these instruments first arrived
And the point of My Dog Has Fleas is to be able to tune and play your Uke without even knowing what GCEA means. That's how Hawaiians played Guitar as well, by ear and sound and memory and feeling and heart.
Even today, most Hawaiians learn how to play the Uke without knowing the names of chords or strings.
Thus, "gifts come every afternoon" has no use in Hawaiian ukulele learning or playing.
My dog has fleas, by contrast, is a melodic saying corresponding to the notes of a song, when sung, and all one needs to do in order to play and sing the ukulele is learn how to sing "My Dog Has Fleas", as once you learn a song you can easily sing it the exact same way every time you open your mouth, wherever you are, making your ukulele easily tunable.
But the trick is someone has to teach you what "My Dog Has Fleas" sounds like.
And the fact that phrase was chosen specifically points quite clearly to the well-known relationship between that and the jumping flea ukulele
On another note, almost everything ever spoken or named by Hawaiians, especially in ancient times and these early post contact days, has Kauna, which is to say that almost everything is a Double Entendre. They were actually scary genius when it comes to Communication and Music. Kauna was necessity, due to the strict kapu laws they lived under that had very harsh punishments, as well as for social expectations and cultural norms, and also for the various beings of the Spirit World, many of whom listen and take action based on what's spoken, and for the existence of extremely powerful type of sorcerers and like witches who practiced a very potent and deadly form of black magic.
So, it's actually very highly probable that the name Ukulele was selected to have one meaning that's proper and royaly appropriate, whilst having a second meaning that's irrelevant and silly or naughty. And for all a Queen would know, it had one meaning, but to common people it had another