As I sketched out in this answer, Ahaz's refusal to ask for a sign is critical to interpreting what follows. For what follows is not best conceived in context as a sign "designed to signify to Ahaz that the land will be delivered from its calamities." Rather, it is indeed a prophecy of judgment.

Ahaz was already told that the Syrian/Ephraim alliance would fail (Isa 7:7-9). When the LORD has Isaiah ask Ahaz to indicate a sign (v.10-11), Ahaz's refusal to (v.12) is not looked upon with favor. Verse 13 is then a key shift in the narrative to the prophecy (NKJV):

Then he said, “Hear now, O house of David! Is it a small thing for you to weary men, but will you weary my God also?

Two things are critical to see:

1. Ahaz's refusal to do what the LORD asked of him is wearying to God, as much of the behavior of the lineage of David had been (both the "you" references in the verse are plural, not singular, in the Hebrew). So Ahaz is being addressed, but obliquely in his connection to the lineage of David, which leads to the second point...
2. Ahaz is no longer the sole referent, but rather the referent expands to the "house of David" (any and all those in the line of David, Ahaz included, but not exclusively or even necessarily to him alone). So the prophecy is for the "house of David" to pay attention to, such that when it is fulfilled, they will take notice of it.

The following prophecy is twofold:

1. A child will be born under specific circumstances (see the previous answer linked to with respect to the virgin aspects, but also below for a key footnote I had there that is really more related to this question [hence why it was a footnote in the other question]), a child whose name implies God is with His people again (Isa 7:14-16)
2. But more immediately, judgment to Judah will be coming (i.e., God not with His people). That judgment is not by the Syria/Ephraim alliance (that was already made clear), but rather by Assyria (Isa 7:17-25), the nation Ahaz would look to for help against the alliance (2 Kings 16), which judgment comes about in Ahaz's son Hezekiah's days (2 Kings 18), though Jerusalem is spared (and Judah for a time) because of Hezekiah's looking/prayer to God (2 Kings 19:20).

As to how the prophecy of Isa 7:14-16 can be conceived as fulfilled in Christ, I'll reproduce from what was my text in footnote 1 in the other answer I linked to above, but fits better in the text of this answer. Everything between the horizontal breaks was originally part of n.1 in that answer.

Note that the only immediate point of v.15-16 is that one land abhorred by Ahaz that has two kings over it will cease to have kings by the time this child of Isa 7:14 is very old. Almost unanimously commentators will identify these two kings with Rezin and Pekah (Remaliah's son), identifying the single land as symbolically illustrating the united forces of two lands (quotes from commentaries found at the preceding link):

The "land" must certainly be that of the two confederate kings, Rezin and Pekah, the Syro-Ephraim-itic land, or Syria and Samaria (Pulpit Commentary)

The countries of Syria and Israel, which Ahaz abhorred for their cruel designs and practices against him (Benson Commentary)

Syria and Samaria regarded as one (2Ki 16:9; 15:30) (Jamieson-Fausset-Brown Bible Commentary)

Ephraim and Syria are treated as one territory, ruled by the two allied kings (Cambridge Bible for Schools and Colleges)

But this is God's message, and though the sign is broadly to the house of David, the last part of v.16 relates it to Ahaz with "The land that thou abhorrest shall be forsaken of both her kings" (KJV). Most later translations have "dread" in place of the idea of abhor, as the word can have either idea (see BDB), and Ahaz's fear is known from v.4.

However, if one understands it as abhor (repugnance) rather than dread (fear), it can be argued that the reference God is making is to the literal singular land of the divided monarchy, that is, Israel and Judah. Ahaz has demonstrated to God his abhorrence for the land in his great wickedness (2 Kg 16:2-4; 2 Ch 28:1-4)—note the various immoral uses of the land in the references, and of course the actions themselves would lead to judgment upon the land. Ahaz's true abhorrence was for the land God had given.

Viewed so, the prophecy fits Christ exactly, for what it is saying is that Israel and Judah (the one divided land, that land abhorred by Ahaz) will both have ceased to have their kings during the early years of the child prophesied in Isa 7:14.

This is not true of Ahaz's day (Judah remained with a king). It is also not true of the time of the Babylonian captivity (no such child was born during that time). But it is true of Christ. A king (Jewish, but not Davidic) was reestablished during the time of the Maccabees in the Hasmonean dynasty:

From 110 BC, with the Seleucid empire disintegrating, the dynasty became fully independent, expanded into the neighbouring regions of Galilee, Iturea, Perea, Idumea and Samaria, and took the title "basileus" [king].

This dynasty was replaced by the Herodian dynasty, with the new king, Herod the Great:

The installation of Herod the Great (an Idumean) as king in 37 BC made Israel a Roman client state and marked the end of the Hasmonean dynasty.

He "was appointed 'King of the Jews' by the Roman Senate," but the more significant point is that:

When Herod died in 4 BCE, the kingdom was divided among his four sons into tetrarchies, the largest being the Tetrarchy of Judea.

The land of Israel was once again without a king, but this time during the youth of the one named Immanuel, Jesus Christ!—

Scholars generally accept a date of [Christ's] birth between 6 and 4 BC. It is generally agreed that Herod the Great died in 4 BCE, placing the birth of Jesus before then.

Then v.17ff goes on to describe the process by which Israel and Judah will begin to lose power (and lose their kings), beginning with Israel (and even Judah is greatly affected by Assyria, so Isaiah 8:8).

Conclusion

Ahaz refused a sign, so he was not specifically given one, but the house of David more broadly was given a sign that God would be with them, the birth of a child named Immanuel under specific circumstances. That sign only and uniquely came to pass at the birth of Jesus Christ, per the statement of Matthew 1:22-23.

A Greek translation of Hebrew almah is parthenos, which can mean both "young woman" as well as "virgin."

There's a bit more detail in this thread.