What is “alliteration” actually?
Solution 1:
Even though the term was coined (by Giovanni Pontano) in the 15th Century, alliteration was already used by ancient Greeks and Romans, in poetry as well as in plain speech.
Caesar's famous phrase
Veni, vidi, vici ("I came, I saw, I conquered")
is a good example.
Thanks to the repeated consonant, a phrase, or a series of phrases, becomes easy to remember, and, in some cases, forces itself upon your memory.
The parallel in music would be a single note played more than once in quick succession that makes the (musical) phrase instantly memorable (think the opening three notes of Beethoven's Symphony Number 5).
Advertising professionals are well aware of alliteration's utility when it comes to forcing something into people's heads (regardless of how folks may feel about it) and have been using the trick for over a century. Consider:
Coca Cola
Dunkin' Donuts
Bed, Bath & Beyond
Best Buy
In the English language at least, consonant alliteration is a great deal more common than vowel alliteration, even though the latter is occasionally used.
The repeating sound (or even a string of sounds) does not have to be the beginning sound of each word being used in alliteration, either. In his poem The Raven, Edgar Allan Poe uses the "ur" sound as follows:
And the silken sad uncertain
rustling of each purple curtain ...
... and reinforces the effect by rhyming the words containing the alliterating sound.
Solution 2:
Fowler is just plain wrong about using letters. Alliteration has to do with sounds only, not letters.
That said, it is true that most alliteration consists of repeating consonants (or more often consonant clusters). This is because there are far more words starting with consonants than with vowels, and far more still starting with consonant clusters than with single consonants.
It is possible to alliterate initial vowels, but it's rare in modern English, because in order to avoid being reduced to schwa, vowels have to be stressed, and most English words beginning with vowels are not stressed on the first syllable.
When vowel alliteration occurs, it is common to precede each alliterated stressed vowel with a glottal stop. German does this automatically for every vowel-initial word, which is part of what makes In Ulm, um Ulm, und um Ulm herum a tongue-twister in German.