There is a notable absence of real names in this book, so character traits have to be formed and determined outside of a name or its meaning. (See, for example, Ofglen's page in the "Characters" section.)
Aside from the professors in the "Historical Notes," it's debatable whether we ever hear a character's real name. The Handmaids are all referred to by names that signify the Commanders they serve: Offred, Ofglen, Ofwarren, Ofcharles, etc. These names are "patronymic, composed of the possessive preposition and the first name of the gentleman in question" and, like the other names in the text, they are "equally useless for the purposes of identification and authentication" (Historical Notes.29, 30).
They reveal nothing about the Handmaids, except the absence of their own identities and personhood. Similarly, even though the Aunts have individual, feminine names, they aren't their real names either. They've been renamed by the administration in references to domestic products (Historical Notes.37).
Even when the narrator refers to other characters from her past by individual or specific names, it doesn't necessarily mean those are their real names. We just don't know. The professors speculate in the "Historical Notes" that the narrator is trying to protect the people she loves by using pseudonyms for them.
However, these pseudonyms may not be arbitrary. "Luke," the name she uses for her husband, could be a Biblical reference to Saint Luke. The name she uses for her best friend, "Moira," is an Irish version of "Mary" and can thus be seen as a reference to the Virgin Mary or Mary Magdalene. At one point, the narrator seems to emphasize both the illusion of these names and their religious connections by asking God to reveal a name too: "I wish you would tell me Your Name, the real one I mean" (30.34). Significantly, the narrator's daughter and mother are never named.
The narrator calls her real name her "shining name" (14.38), which she reveals to Nick but never to us. At least one reviewer believes clues in the text point to it, however. Mary McCarthy observes in the New York Times that "we are never told [the narrator's] own, real name in so many words, but my textual detective work says it is June" (source).
The narrator lists several women's names in the first chapter. We hear all of them again at various points in the book except for the last one, June. If "June" is the narrator's real name, that too has significance: the month of June was named after the Roman goddess Juno, who ruled over marriages.
In Western society we're used to thinking of clothing as a means of expressing our individuality and personal style. What you wear helps reveal who you are, right?
The narrator grew up with this notion, but it was taken away from her when she became a Handmaid. By the time she sees Japanese tourists on the street, she is "fascinated, but also repelled" (5.33). She remembers that "[she] used to dress like that. That was freedom" (5.34). These people are showing who they are by what they wear, and they get to decide who that person is.
In Gilead, the opposite is true. Everyone dresses alike within their social group; clothing reveals status while masking individuality, which is discouraged. The clothing restrictions in Gilead take uniforms to a whole new level of wrongness, pointing to the complete absence of choice. Early in the book the narrator describes the Handmaid outfit she's condemned to wear:
The skirt is ankle-length, full, gathered to a flat yoke that extends over the breasts, the sleeves are full. The white wings too are prescribed issue; they are to keep us from seeing, but also from being seen. I never looked good in red, it's not my color. (2.8)
The clothing the Handmaids wear is supposed to make them all the same, to other people and to each other. Their clothes both blind them to the outside world and keep them hidden from it. The narrator rejects her clothes, even though she has to wear them, by saying red is "not [her] color." Red, of course, is the color of blood, of Communism, of the adulteress' "A" in The Scarlet Letter. It's a reminder of passion... even though the Handmaids aren't supposed to be passionate.
Rita's Martha clothes are also interesting:
[…] her usual Martha's dress, which is dull green, like a surgeon's gown of the time before. The dress is much like mine in shape, long and concealing, but with a bib apron over it and without the white wings and the veil. She puts on the veil to go outside, but nobody much cares who sees the face of a Martha. (2.13)
Rita is made less by her clothes. If she's wearing a green "Martha's dress," no one will be interested in looking at her as an individual person. She's just a servant. The Marthas' dress makes the women who wear it, like Rita and Cora, serviceable but not desirable—useful and invisible.
Women are divided into a small range of social categories, each one signified by a specifically colored dress in a similar style. Handmaids wear red, Marthas wear green, and Wives wear blue. Econowives, the lower-class women who still have minimal agency, are sort of a mixture of all these categories, so they wear stripes (5.5).
The dress code means that women are no longer individuals; they have become interchangeable, like identical outfits on a rack. The narrator realizes the consequence of this in one of her conversations with the Commander, when she says, "we don't have different clothes [...] you merely have different women" (37.26). (For more on this, see the "Identity" and "Women and Femininity" themes.)