Carnegie is citing a quote from Hans Selye, but I'm inclined to think it doesn't have the exact sense of most statements using this well-established general format, which are normally along the lines of...
"As much as I love you, I can't leave my husband"
"As much as we want to, we just can't get married"
...where As much as X, Y essentially means "Even though X is true/desirable, it is incompatible with Y. It's normally used in contexts with the strong additional implicit meaning Y is the overriding factor.
Selye was a Hungarian endocrinologist, much concerned with biological stress (which had not previously been significantly studied, or even recognised). Taking into account that background information, and Carnegie's likely intention in quoting the words, I think OP's quote is probably best interpreted literally, as simply saying our fear of condemnation and our desire for approval are equally strong conflicting urges.
That's to say, in this particular case, as much as probably really does mean to the same extent that, but normally it's a "set format" construction where the second "statement of fact" overrides the first "desire".
Noting OP's amendment that the quote was given in a context where Carnegie's main point was that "criticism is a bad idea", and the fact that it's in a book supposed to teach you how to win friends and influence people, I imagine he's advising against constructive criticism. That's popularly interpreted as being where you explain what people are doing wrong (so they can change, and thus improve). But you make more friends by ignoring what they're doing wrong, and praising what they do right.