In the passage, the meaning of "alliteration" is the repetition of consonants at the beginning of two or more words next to each other, and sometimes separated by short words like "to" and "and". We can infer this because of the examples the author provides after the word "alliteration": 'British brethren,' 'time to time,' 'common kindred,' 'which would,' and 'connections and correspondence' which are all words that begin with the same consonant sound.