I think that oddly enough the language features that Lewis Carroll uses in this poem are designed to obfuscate any meaning rather than create it. It is, after all, meant to be a nonsensical poem.
For example, the poem is dense with a range of pronouns (e.g., they, me, you, her, him, she) which are deliberately piled one of top of the other to make it very difficult, if not impossible, to tell with any clarity or confidence exactly who each pronoun refers to and what the relationships might be between all these people. There are four different pronouns in the very first line. That's an indication from Carroll that anybody hoping to make sense of the rest of the poem should abandon all hope. Indeed, Alice exclaims, upon hearing the poem, "I don’t believe there’s an atom of meaning in it."
There is also in the poem quite a strict rhyme scheme, with alternating end-rhymes throughout the whole poem ("her/him/character/swim") and also a regular syllabic meter, with every odd numbered line having eight syllables, and every even numbered line having six syllables. These features create a strong rhythm, which in turn lends to the language of the poem a repetitive, sing-song tone, compounding the impression that the language itself is childish nonsense.