"Tithonus" is a dramatic monologue spoken by the title figure of myth, but in my view, the term "elegiac" does not precisely apply. An elegy is more specifically a poem for the dead, such as Milton's "Lycidas" or Shelley's "Adonais." Tithonus is lamenting not death, but conversely, his own immortality. The poem carries the tone of grief, as an elegy does, but it's even sadder than a true elegy, because the message of an elegy is that the dead have found a better life, a kind of glory, or simply rest beyond the grave. Tithonus has none of these things, because he is condemned to live for ever.
In mythology, Tithonus is granted eternal life by Eos, goddess of dawn. Hence the repeated allusions by Tennyson to "the East" and to the love which Tithonus and the goddess once had:
I used to watch—if I be he that watch'd—
The lucid outline forming round thee; saw
The dim curls kindle into sunny rings;
Changed with thy mystic change, and felt my blood
Glow with the glow that slowly crimson'd all
Thy presence and thy portals, while I lay,
Mouth, forehead, eyelids, growing dewy-warm
With kisses balmier than half-opening buds
Tithonus, though immortal, has not retained his youthfulness. Though the fact that he is now a "gray shadow, once a man" is the obvious reason for his lament, there is another factor here. In this, and other poems by Tennyson and his contemporaries, we sense an existential weariness about life itself, a desire to escape it and find peace or oblivion. In Swinburne's "The Garden of Proserpine," the speaker tells us that
We thank with brief thanksgiving
Whatever gods may be
That no life lives for ever;
That dead men rise up never;
To me, it is interesting that the Victorians were considered dated, old-fashioned, and irrelevant by the modernist generation of T.S. Eliot, Ezra Pound, and the others, but much of the message of Tennyson, Swinburne, Matthew Arnold and others is similar to that of the twentieth-century poets. "Tithonus" anticipates the tone of Eliot's "Gerontion." Tithonus himself, in Tennyson's hands, becomes a symbol of man alone in the universe and wishing to be released from life.