There is a common expression "to give (someone) the clothes off your back", meaning that you would go without basic necessities yourself in order to provide for that person; the Dursleys certainly haven't been doing that for Harry! JK Rowling is playing with our (the readers') presumed understanding of that expression.
His uncle is using "clothes" figuratively, as a stand-in for all the necessities of life; Harry's answer turns this around - his clothes are literally Dudley's hand-me-downs, but presumably the food he eats hasn't gone through Dudley first!
Side note: Clothes have a lot of significance in the Harry Potter books; house elves are automatically freed (and banished) if their owners give them clothes. Coincidence?
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.
Best Answer
The saying born and bred dates back at least to the 17th century as shown in Ngram. To breed at that time already meant also to grow up ( late 14th c.) so there is not reason to suppose that the expression had originally a meaning different from the contemporary one.
Born and bred:
Breed (v.): (Etymonline)