Steven Pinker does not say that “can’t get any” is a double negative,
merely that it is no less of one than “can’t get no”.
I think he is exaggerating, to make the point that grammatical rules are arbitrary rather than logical.
What Pinker actually says
The questioner writes
I don't understand why "can't get any" is a double-negative.
Steven Pinker does not claim that it is a double negative, only that
‘“can’t” and “any” is just as much of a double negative as “can’t” and “no”.’
What I think he means
I take him to mean that English works equally well whether the rule for totally negating “can get” is to say “can’t get no” or to say “can’t get any”, and he goes on to point out that it is a historical accident that “can’t … any” came to be preferred by those laying down the rules.
He is explaining the difference between descriptive and prescriptive rules, and says that
many of the prescriptive rules make no sense whatsoever
and uses the rules forbidding “split infinitives” and “double negatives” to illustrate this.
He is making the point that prescriptive grammar is more arbitrary and less logically compelling than people often suppose it to be, and that dialects which do not conform to it can be just as complex and expressive.
I do not, however, think it fair to say (as he does not explicitly) that the rule forbidding double negatives makes no sense whatsoever; in fact it codifies a convention which may be useful in some contexts.
We can define “double negative” so that the rule makes sense
What I think is confusing about Pinker’s claim is that “no” is in fact considerably more strongly associated with negation than “any” is. If we choose to therefore classify “can’t” and “no” as negative, but not “any”, and define a “double negative” as a construct using two words we classify as negative, then “can’t get no” is a double negative while “can’t get any” is not.
The rule does not forbid all double negatives and may be useful
Moreover, what the rule actually means is not to avoid a double negative, but to avoid using it to express simple negation: there are two negatives in “don’t say he’s not welcome”, “say not the struggle nought availeth”, “let nobody say you are not a sensualist” and perhaps even “thou shalt not bear false witness”, but all are acceptable. The point is that these are interpreted so that one negative negates part of the utterance containing another negative, while the negatives in “can’t get no” support one another.
If we want to construct complicated sentences, perhaps in a mathematical or philosophical context, in which we can negate arbitrary constructs, we need to agree when one negative negates a construct, and when two negatives belong together to form a simple negation. A simple rule to achieve this is to forbid the latter case – but this comes at the expense of classifying many common utterances as incorrect.