This is a question that may cause Rabbi Adelson to gnash his teeth. The answer is a qualified “no.” (Rabbi Adelson is now jumping up and down saying, “It is not qualified.” Let me explain, please, Rabbi.)
Bat mitzvah as a term for a young woman observing a rite of passage is a more recent term meant to be equivalent to that of a young man becoming a bar mitzvah. The term we should really look at is “bar mitzvah.”
The word “bar” is Aramaic, and is, on its face, basically equivalent to “ben” in Hebrew. Names were called back in the Aramaic day along the line of “Simon bar Kosiba” (which was Simon bar Kokhba’s original name, back in the second century CE). Nowadays we use “ben.” (It sometimes looks on our gravestones as if we are writing “bar,” but that is an abbreviation, with a geresh between the bet and resh, for “ben reb” or “son of the honorable.”)
But also going back to that day, we find a sort of idiom which is really not translatable. Folks would refer to someone as a “bar XXXXX” to mean, roughly, a “dude who is into XXXXX,” or an “XXXXX guy.” Rabbi Adelson points out the Talmudic term “bar plugta.” “Plugta” is Aramaic for “disagreement.” A “bar plugta” is “one who is generally in disagreement with another specific person.” Those rabbinic cohorts whose commentary we read in the Talmud may have been friends, but often they were “bar plugta” of one another.
Rabbi Adelson also points out “bar da’at,” which is “one who possesses knowledge,” since “da’at” is “knowledge.” And Simon bar Kokhba’s name? “Kokhba” means “star,” a name said to have been given him by Rabbi Akiva, to refer to Jacob and the star prophecy (Numbers 24:17). Simon the Star Guy.
In researching this idiom, I found much discussion by our Christian friends, concerning their leader referring to himself as “son of man” or “son of humanity” (“bar nasha” or “bar enasha”). Not unlike “ben adam” in Hebrew which is found in several places in Tanakh and means literally “a human being,” it might have meant he was just a guy, or was way into humanity. In a Cambridge University publication of 2012, scholar Delbert Burkett posits (for a publication) three possible idiomatic senses of that expression: the circumlocutional sense (the humble “this man”), the generic sense (this guy, myself), and the indefinite sense (a man, someone). But at least in his extract he fails to mention the idiom we discuss above, though it could be included in the circumlocutional sense. (Your humble correspondent is sometimes a bat circumlocution, get to the point already.)
In recent times, we have used the term “Flower Children” in a similar way; they pointedly were not children of flowers.
So Rabbi Adelson suggests that since the literal is impossible (“son of commandment”) we translate the term (if we must) as “one who has reached the age of majority and therefore is subject to the mitzvot.” As my mom (z”l) sometimes exclaimed, “son of a seacook!”