The two houses, that of Jacob and that of Esau, are representative of the two great classes of people: the righteous, and the wicked.
The "house of Jacob" refers to the Israelites, for "Israel" was the new name which God gave Jacob following his night of wrestling with God as he was in fear of Esau.
The Bible indicates that we may become part of "Israel" by accepting the truth. All of the covenant promises are for "Israel"--including salvation. For example, consider this:
And so all Israel shall be saved: as it is written, There shall come
out of Sion the Deliverer, and shall turn away ungodliness from Jacob:
(Romans 11:26, KJV)
But that this does not mean literally all those who are born as Israelites is made plain two chapters earlier.
Not as though the word of God hath taken none effect. For they are not
all Israel, which are of Israel: (Romans 9:6, KJV)
In other words, "Israel" is symbolic of the righteous. This is why Revelation addresses the saved as coming from the twelve tribes of "Israel"--with nary a mention of any non-Israelite tribe or group of people in heaven.
The "house of Esau" refers to those who are not part of Jacob, or "Israel." We might remember the text where these two classes are compared, which says:
As it is written, Jacob have I loved, but Esau have I hated. (Romans
Again, "Esau" stands for the great class of the wicked here. God is not saying He will accept "Israel" and reject Edom, Moab, Ammon, Babylon, Ninevah, etc. In that single name "Esau", all those who are among the unsaved are represented.
Will every single person of this "house of Esau" be destroyed? Yes. Only the righteous will be permitted to live in the consuming fire of God's presence, the "everlasting burnings" according to Isaiah 33:14 (cf. Isaiah 43:2 & Psalm 15) in the new earth. The wicked, represented in Obadiah as "the house of Esau," will all be destroyed.