The first prophecy is centred on Ahaz, the second is interpreted by Jews to mean King Hezekiah. They appear to refer to completely different time-periods, and there is certainly nothing to link them apart from being prophecies relating to the birth of children, the second one certainly appearing to have extraordinary characteristics. The first Isaiah certainly seems more concerned with events and conditions immediately present or in the not so distant future. One wonders what Jews were to make of a prophecy whose fulfilment was to occur - what? more than 500 years later.
Some Jewish teachers said that Is. 9:6-7 can't be about Hezekiah.
If the first prophecy does refer to a virgin who is currently pregnant, then that would be a miraculous sign, which would link it with the 'extraordinary characteristics' of the child in 9:6-7.
(The usage in Song of Solomon 6:8, which I think talks about the women of the king's harem "There are sixty queens and eighty concubines, and virgins, wa·‘ă·lā·mō·wṯ, without number" seems to point to a definition of almah as a woman who has not yet slept with a man).
I came across in an old study bible a note that I wrote (not sure where I got it from), to the effect that Immanuel in 7:15 lives on milk and honey, which represents the remnant learning not to sin. They live in poverty until they know to reject the evil and choose the good. This ties in with 7:21-22, "In that day a man will keep alive a young cow and two sheep, 22and because of the abundance of milk that they give, he will eat curds, for everyone who is left in the land will eat curds and honey."
I also found the following in
Meyer's commentary on Matthew 1:23"With greater weight and clearness Kahnis (Dogmatik, I. p. 345 f.) remarks: The Virgin and Immanuel are definite but ideal persons. The latter is the Israel of the future according to its ideal side; the Virgin, the Israel of the present and of the past according to its ideal side, in accordance with which its vocation is, by virtue of the Spirit of God, to give birth to the holy seed; this Israel will one day come to its true realization in a virgin, who will be the mother of the Messiah."