The usual, humorous, phrase is like this:
My daughter is 16, going on 32!
It simply means she is precocious. She is only 16, but she already behaves in a very mature way. So in the usual phrase, the age difference goes upwards. Another example, "My kindergarten kid is 5 going on 10!"
That's the usual direction of the joke. But here, the author here is deliberately turning it around the other way.
So indeed the author is expressing that the woman is 22 but behaves immaturely.
To understand this usage:
In English commercial writing, in the present day, there is a fad to take an existing humorous phrase, and "turn it around". The idea is that it (supposedly) sounds even more witty when reversed. You could say this is an "overused trick" in English commercial writing today. The example at hand is precisely an example of that process.
(Note: as Robusto explains, "going on" very simply means "almost". For example, "to walk to the store is five, going on six, miles", "renovation costs are 80 thousand, going on 90 thousand.")
So, to get the entire feel of the passage in English relies on the following chain:
1) "Going on" means "almost": the child is six going on seven. That sentence simply means "almost seven".
2) Very commonplace humorous use of "going on" with a large gap going upwards, used specifically of precocious children: that girl is 15 going on 35!
3) In this case, the author has "turned around" that usual humorous pattern: "the person is 35 going on 15". Note again that it is common (today) in commercial English to invert a common humorous construction, to create a (supposedly) even funnier one.
By the way, the phrase Sixteen Going on Seventeen is indeed one of the handful of most famous "showtunes" in all of English, 1965,
https://www.youtube.com/watch?v=hwK_WOXjfc0
(immortal performance in the film by Charmian Carr) So that song was written by Oscar Hammerstein II and Richard Rodgers (the two most famous "showtunes" era composers) and it's one of the songs in The Sound of Music (far and away the most popular musical show and film in English).
So, for any English speaker, whenever you say or hear the phrase "16 going on 17" (much as with "do-a-deer", "brown paper packages" "edelweiss" and indeed others from the same show) it associates instantly with the song.
I disagree with JLG's explanation. Your supervisor is conflating two similar-looking but non-identical constructions having two different meanings:
1) To conceive of X as Y, where conceive means to interpret / comprehend:
Copernicus conceived of the Earth as a sphere rather than as a plane
2) To conceive X as Y, where conceive means to invent, especially when the invention is in pursuit of a particular objective:
Julius Edgar Lilienfeld conceived the field-effect transistor as a solid-state replacement for the triode.
Best Answer
Firstly, this isn't a specifically British construction, nor is it the standard in British English. It's just a nonstandard variant that occasionally occurs in both American and British English. (More on that later below.)
As for an explanation of why it is used, I guess some people either understand the word "miss" differently than we do, or just aren't thinking clearly.
A different understanding of "miss":
"I miss not having him around" seems like a cross between "I miss him" and "I regret not having him around". The standard meaning of "I miss X" (in our context) is something like "I feel a sense of absence or loss because X used to be around and isn't any longer", thus "I miss having him around". Some people appear to be extending/inverting it to "I feel a sense or absence or loss because of X" (see the quote *"I miss that he is not around any more" in the other answer) thus for these speakers, "I miss not having him around" means "I feel a sense of something missing because of not having him around."
You can see this most clearly in the American examples:
where there is talk of missing things that one never had, and
Here the question uses "miss" in the nonstandard sense ("do you feel something missing because of not having hands and feet", a rather daft question), while the answer, having ignored the not (and the literal meaning), treats "miss" in the standard sense: I can't miss something I've never had.
As for why this different sense comes about, I guess there's a strong temptation, in phrases involving counterfactuals, to state the factual instead. (I suspect something similar was at work in the origin of "could care less", sometimes claimed to be sarcastic though it's not.)
Not thinking clearly:
This actually happens naturally in language. It's not unique to this phrase; there's a small amount of confused inversion in many phrases involving negatives:
I guess that, with time, "miss not" will join the list of phrases like "could care less", "doesn't know squat", and "teach you to", where a negation can be added or removed with no change in meaning. So this inverted usage of "miss" may well become standard, and it will only be considered to add to the glorious irrationalities of the English language. (In the Corpus of Historical American English, usages of "miss not" before 1968 are mostly the "correct" one, while 1968 and later has mostly the usage you asked about in the question.)
Usage patterns
Corpora indicate it may even occur more often in American English than British English, though more below.
The British National Corpus has only five instances of this usage, all from fiction: "we miss not having your funny face around to laugh at", "I miss not having one", "D'you miss not drinking very much?", "Did you miss not 'aving me to talk to?", and one that, interestingly enough, uses both the "miss not" and "miss" constructions in a single sentence:
The Corpus of Contemporary American English, however, has 20 instances of such usage (leaving out phrases like "miss not only…"), from news, spoken TV programmes, magazines, fiction, etc., e.g.: "BH: Do you miss not having children? Calhoun: Yes. You know I love kids", "Do you ever miss not having a daddy?", "I will miss not being able to walk Christie down the aisle", "I miss not having the late Red Auerbach to answer such vital questions", "I miss not being able to talk to people in the market", "I miss not being in a more cosmopolitan city", "'Do you miss not having hands and feet?' Well, I've never had them.", "I miss not being able to see my friends' faces", "I miss not having my dog around", "I feel that as an only child she may miss not having siblings later on, as an adult.", "I kind of miss not being able to drop by my parents' house during the week", "Cmdr. VERNOSKI: Listen, do you miss not having my long hair on the rug? Col. VERNOSKI: Absolutely! I miss not having your long hair in my hands."
It does contain one "correct" usage of "miss not" as you and I seem to understand it:
Of course, probably the two corpora aren't the same size, but I don't (from looking at them) see any clear justification for calling it a British construction. And from the relatively small number of instances, it seems less frequent than the "miss having…" variant.
Another way of looking at them is through Google n-gram viewer. I don't think this is extremely reliable, but worth looking.
Here are "miss having" and "miss not having" in American English:
And in British English:
Similarly, here are "miss being" and "miss not being" in American English:
And in British English:
So from the data it seems (look at the numbers on the y-axis carefully) that "miss not being" and "miss not having" are used at about the same frequency in both American and British English. But "miss having/being" are used much more in American English, so the "miss not" forms are more common in British English relative to the standard forms. Still, less frequent. (This is a crude search without context, so all the caveats apply.)
Edit [2011-11-23]: The Language Log has had around 75 posts on this very topic. It has been enlightening reading several of them, but I am too exhausted to summarise them here, and I recommend you to read those posts directly.