To his amazement the girl did kiss her hand and stretch it out.
In my understanding, to mostly refer the object of an action. So what’s the meaning of to in this sentence?
I think, the sentence should be rewrite to something like the following sentence.
Amazemently, the girl did kiss her hand and stretch it out.
The sixth entry in the New Oxford American Dictionary’s listing for to is:
to 6 governing a phrase expressing someone’s reaction to something : to her astonishment, he smiled.
So to in your example is simply a marker to introduce the reaction and identify the person having the reaction: To whose amazement? To his amazement. (Whoever “he” is.)
(And as others have suggested, you really never want to use bizarre coinages like amazemently in a sentence.)