Indeed, Rashi (Kesubos 8a) agrees with your intution:
והתקין לו ממנו: מגופו מצלעותיו
And he fashioned for him, from him: From his body; from his ribs.
However, an alternate explanation is noted by R. Aaron ben Jacob Hakohen of Lunel in Orchos Chaim (Hil. Kiddushin § 21):
והתקין לו ממנו בנין עדי עד: פי׳ הנפש השכלית
שנחצבת מגלגל תשיעי ועומדת לעד בקיומה שלא תכלה
And he fashioned for him, from him, an everlasting building: Meaning the intellectual soul, carved from the ninth orbit and remaining forever in its state of existence so that it will not end.
According to this explanation, "building" refers not to Eve, but to the soul. Accordingly, "from him" presumably means from God, because presumably God did not make a soul for Adam, from Adam, and God is the only other entity mentioned in the blessing that "him" could refer to.
Thus, the siddur in question may follow this interpretation.
If I might add, this interpretation seems a bit far fetched, considering that the term "building" is explicitly employed by the biblical text in reference to the creation of Eve (Genesis 2:22) and not in reference to the soul (see Genesis 2:7).