The first verse mentioned in the OP headline is this:
Fear not, for I am with you;
from the east I will bring back your offspring,
from the west I will gather you. (43:5)
Cyrus is mentioned specifically in the next chapter as God's instrument to liberate Judah from Babylonian oppression:
I say of Cyrus, My shepherd!
He carries out my every wish, Saying of Jerusalem, “Let it be rebuilt,”
and of the temple, “Lay its foundations.” (44:28)
Then, in chapter 46, as mentioned in the OP, the prophet predicts that the gods of Babylon will bow, because
I summon from the east a bird of prey,
from a distant land, one to carry out my plan. Yes, I have spoken, I will accomplish it;
I have planned it, and I will do it. (46:11)
How can we know this refers to Cyrus? The submission of the gods of Babylon to him is one clue. But if we back up to chapter 41 we find still another reference to a military leader from the east who will liberate God's people.
Keep silence before me, O coastlands;
let the nations renew their strength. Let them draw near and speak;
let us come together for judgment. 2 Who has stirred up from the East the champion of justice,
and summoned him to be his attendant? To him he delivers nations
and subdues kings; (41:1-2)
If these chapters are read together, it's clear that Cyrus (who is named in chapter 44) is the same person - God's champion from the east - who liberates God's people in chapters 41, 43 and 46.
However, this does not prevent people from understanding the prophecy to refer to the current era. A similar principle applies to Isaiah 7-8, where the prophecy of Immanuel was fulfilled in Isaiah's day, but is later understood by Matthew's Gospel to apply to Jesus. Similarly, the author of Revelation may have been thinking of Nero Caesar as the anti-Christ, but almost every generation has come up with their own candidate.
Of course it is highly unlikely for people who live elsewhere to think that Isaiah refers to the Philippines. But in principle, a scripture that applies to an ancient time is sometimes sincerely believed to apply to the present era as well.