$ man bash
Word splitting and filename expansion are not performed on the words between the ‘[[’ and ‘]]’; tilde expansion, paramete
This is not the behavior I would have expected either. However, I do not believe it is due to the man page entry you've cited, but rather due to the behavior of =~.
My guess is that " is interpreted as a literal character in the Extended Regular Expression.
For example,
[[ hello = "hello" ]] && echo YES || echo NO
YES
So the double quotes are ordinarily stripped.
Consider also grep, on the shell:
echo foo | grep '"foo"' && echo YES || echo NO
Versus:
echo foo | grep "foo" && echo YES || echo NO
foo
YES
In this case, "s are removed by the shell before grep receives them. In the latter case, grep receives the quote, and the regular expression engine determines it not to be a match.
I posit that precisely this is the case for =~.