It's just a phonetic loan (A-mei-rica) represented by a character with positive associations (they could also have chosen e.g. 没国 "without country", but that would've been impolite). Best not to read too much into it.
England is called the country of heroes, France the country of law, Germany the country of virtue... people rarely pay attention to the literal meaning.
Historically, going abroad means wealth and status, marrying whites ("ocean people") too, those who return with immense wealth and fanfare reinforce this notion, and it just so happens that the transliteration of "me" was the Chinese word "mei".
