If someone is listening to another person praying out loud, and they respond, "amen," does Hashem consider the response to be a prayer in its own right?
Hashem does consider it a prayer, and it is considered to be the same prayer recited by petitioner.
Whenever a person listens to the entire recitation of a blessing with the intention of fulfilling his obligation, he is considered to have fulfilled his obligation although he does not answer Amen. Whoever answers Amen to a blessing recited by another person is considered as if he recited the blessing himself, provided the person who recites the blessing is obligated to recite that blessing.
Rambam, M"T, Hil. Berakhoth 1:11
An extension of what you refer to is the Shome'ah K'oneh (One who listens is considered as if he answered.) principle. This is mentioned in Bab. Talmud Succah 38b.
It is generally suggested that a person who answers Amen is considered as if he said the blessing. This principle is used in mitzvoth such as Megillah reading and listening to the shofar. (Note in particular regarding Shofar, that the shofar blower says in the blessing "To hear the sound of the shofar." This, obviously, includes himself.)
The Talmud page (see also Rash"i as well as Tosfot commentary, there) state that even if one merely listened but paid attention to the prayer / blessings, and did not respond is considered as if he prayed or said those items. The example they cite is if someone is in the middle of his silent amidah and he hears the shat"z saying kedusha. He cannot answer kedusha with them. Therefore, if he stopped his prayers, remained silent, listened and paid attention, it is considered as if he said kedusha, himself.