It appears that the Church's typological approach to interpreting Messianic prophecy is grounded for the most part in the New Testament information that is later referenced to Old Testament passages. What is primarily considered to be 'Messianic' in nature is largely attributed to that which refers, in hindsight, to Christ, who is in fact the Messiah. The Jewish considerations throughout the years of these same passages as prophecy is a secondary, though important point as well. A helpful link that includes a somewhat extensive and easy to access list of the New Testament verses that either reference or can be referenced to particular Old Testament passages that are typological in nature can be found here. The following will address the latter point, in regards to whether Jewish culture likewise used typology and if so in what way.
The question to be answered in addition to the Christian typology is naturally whether the Jewish culture around this time both approached Scripture typologically and applied such an approach to their own understanding of what was to come. We could of course refer to Scripture itself, which appears to maintain a typological approach within its own text. The following are a few examples:
Unless the LORD Almighty had left us some survivors, we would have become like Sodom, we would have been like Gomorra. Hear the word of the LORD, you rulers of Sodom; listen to the law of our God, you people of Gomorra! (Isaiah 1.9)
Go now to the place in Shiloh where I first made a dwelling for my Name, and see what I did to it because of the wickedness of my people Israel. While you were doing all these things, declares the LORD, I spoke to you again and again, but you did not listen; I called you, but you did not answer. Therefore, what I did to Shiloh I will now do to the house that bears my Name, the temple you trust in, the place I gave to you and your fathers. I will thrust you from my presence, just as I did all your brothers, the people of Ephraim. (Jeremiah 23.14)
Everyone who quotes proverbs will quote this proverb about you: "Like mother, like daughter." You are a true daughter of your mother, who despised her husband and her children; and you are a true sister of your sisters, who despised their husbands and their children. Your mother was a Hittite and your father an Amorite. Your older sister was Samaria, who lived to the north of you with her daughters; and your younger sister, who lived to the south of you with her daughters, was Sodom. You not only walked in their ways and copied their detestable practices, but in all your ways you soon became more depraved than they. (Ezek 16.44)
Isaac prayed to the LORD on behalf of his wife, because she was barren. The LORD answered his prayer, and his wife Rebekah became pregnant. The babies jostled each other within her, and she said, "Why is this happening to me?" So she went to inquire of the LORD.
The LORD said to her, "Two nations are in your womb, and two peoples from within you will be separated; one people will be stronger than the other, and the older will serve the younger." (Genesis 25)
The LORD will dry up the gulf of the Egyptian sea; with a scorching wind he will sweep his hand over the Euphrates River. He will break it up into seven streams so that men can cross over in sandals. There will be a highway for the remnant of his people that is left from Assyria, as there was for Israel when they came up from Egypt. (Exodus 14.6)
As in the days when you came out of Egypt, I will show them my wonders (Micah 7.13)
All of these verses show both examples of interpreting present situations in terms of past events and predictive cases in which a certain 'type' is indicative of what will occur in the future. Thus, one commentator states "the data from the OT shows quite clearly that typology was a mainstream interpretive approach to Israel's history , personages, events and primeval history. Accordingly, NT writers were not employing an 'alien' method to the exegesis of the OT-- indeed, they were squarely in the mainstream of accepted hermenuetical approach." This indicates that what often appears to be simply historical reference (as certain Messianic prophecies are) can be taken to be, or implied to be in the writing itself, another thing entirely.
Though an objective fact itself, it can also be written in hyperbole, or talking about a very public and/or mystic figure, by which a future tense and further meaning is meant to be taken. These passages can be identified, if they are Messianic especially, as referring to a mystical king or one who is meant to restore some truth and balance. Thus, in the Messianic prophecies we have, the Church is employing an approach very common to Jewish understanding about passages that seem to qualify as speaking in a language of 'types'. This of course does not make these passages 'prophetic' however. It should be noted though that prophecies can very well be formulated in a typological way, meaning that in speaking about a certain historical king, a prophet can stress certain truths that shall occur to a kingly figure beyond the king being apparently spoken about in the actual text. This is very evident in the many passages of prophecy regarding King David and the relation to the Messiah.