The origin of this term is tannaitic and is found in a beraisa that is quoted on both Yevamos 62b-63a and on Sanhedrin 76b (text used here is from MS Munich 95 for the Yevamos text):
האוהב את אשתו כגופו, והמכבדה יותר מגופו, והמדריך בניו ובנותיו בדרך ישר', והמשיאן סמוך לפירקן - עליו הכתו' או' "וידעת כי שלום אהלך."
האוהב את שכיניו, והמקרב את קרוביו, והנושא את בת אחותו, והמלוה סלע לעני בשעת דוחקו - עליו הכתו' או' "אז תקרא ויי יענה תשוע ויאמר הנני."
One who loves his wife like himself, and one who honors her more than himself, and one who guides his sons and daughters on a proper path, and one who marries them off just directly prior to their reaching puberty - upon him Scripture says "And you will know that your tent is peace" (Job 5.24).Both of the two halves of this text each start out with "one who loves", has four actions, and has a Biblical verse speaking to the meritoriousness of the actions. For our purposes here, we will focus on the first half of this text. The first two pieces are about a husband's actions and the second two are about a father's actions. Achieving to love his wife like himself and to honor her more than himself are no easy tasks.
One who loves his neighbors, and one who draws close his relatives, and one who marries his sister's daughter, and one who lends a sela to a poor person in the hour of his need - upon him Scripture says "Then you will call and God will answer; you will cry and He will say, 'Here I am'" (Isaiah 58.9).
However, from the above text, there is no comparison being made between a husband and a wife, except that, for a man to love his wife like himself, he is reached a special level, so to speak/type.
The other two uses of the phrase "his wife is like himself" are each used by the stammaim (the later, redactorial layer of the Talmud). One use is found on Berakhos 24a and the other on Bekhoros 35b. Both of them are found within discussions, though I've tried to excerpt some of the discussions, so as not to be too lengthy.
Here is the Berakhos text (from MS Munich 95):
The Bekhoros text (from MS Vatican 120):
בעא מיניה רב יוסף בריה דרב נחמיה מרב יהודה: "שנים שהיו ישנים במטה אחת - מהו שיחזיר פניו ויקרא ק"ש ויחזיר זה פניו ויקרא ק"ש?"
א"ל הכי אמ' שמואל: “מותר, אפי' אשתו עמו."
מתקיף לה רב יוסף ולא מיבעיא אחר.
אדרבה: אשתו כגופו, אחר לאו כגופו.
Rav Yosef, son of Rav Nehemiah, inquired of Rav Yehudah: “Two people who are sleeping in one bed – should one turn his face and read out the reading of the Shema’ and the other turn his face and read out the reading of the Shema’?”
He said to him: “Shmuel said: ‘It is permissible – even if his wife is with him.’”
Rav Yosef pointed out a difficulty: And it’s not necessary to say another.On the contrary: his wife is like himself – another is not like himself!
ודוקא בנו ובתו, אבל אשתו – לא.
אשתו כגופו דמיא.
And specifically his son or his daughter, but his wife – no.
What is the reason?
His wife is compared as himself.
In these two latter texts (yes, I know there are gaps between each of the lines - if someone knows how to fix that, please let me know), the stam borrows the language from the beraisa out of its context into a different context that suits his needs within a given discussion. However, that isn't to suggest that the stam necessarily misunderstood or misread the original term/phrase, but rather creatively appropriated it for his/their uses in an argumentative framework.
I hope you enjoyed. I think the stammaitic usage of the term is, however, a "fun" way to employ the term, even if not everybody appreciates or accepts its employment.