House of God, in this context = a local assembly, and by extension the Body of Christ.
House of Israel/Judah = Jews only. Not you. Not us.
Both belong to God but are not the same entity, nor interchangeable.
Contexts.
There is only one House of God. not two.
Heb 3:1
Therefore, holy brethren, partakers of the heavenly calling, consider the Apostle and High Priest of our confession, Christ Jesus,
Heb 3:2 who was faithful to Him who appointed Him, as Moses also was faithful in all His house.
Heb 3:3 For this One has been counted worthy of more glory than Moses, inasmuch as He who built the house has more honor than the house.
Heb 3:4 For every house is built by someone, but He who built all things is God.
Heb 3:5 And Moses indeed was faithful in all His house as a servant, for a testimony of those things which would be spoken afterward,
Heb 3:6 but Christ as a Son over His own house, whose house we are if we hold fast the confidence and the rejoicing of the hope firm to the end.
Heb 3:7 Therefore, as the Holy Spirit says: "TODAY, IF YOU WILL HEAR HIS VOICE,
Heb 3:8 DO NOT HARDEN YOUR HEARTS AS IN THE REBELLION, IN THE DAY OF TRIAL IN THE WILDERNESS,
Heb 3:9 WHERE YOUR FATHERS TESTED ME, TRIED ME, AND SAW MY WORKS FORTY YEARS.
Heb 3:10 THEREFORE I WAS ANGRY WITH THAT GENERATION, AND SAID, 'THEY ALWAYS GO ASTRAY IN THEIR HEART, AND THEY HAVE NOT KNOWN MY WAYS.'
Heb 3:11 SO I SWORE IN MY WRATH, 'THEY SHALL NOT ENTER MY REST.' "
Heb 3:12 Beware, brethren, lest there be in any of you an evil heart of unbelief in departing from the living God;
Heb 3:13 but exhort one another daily, while it is called "TODAY," lest any of you be hardened through the deceitfulness of sin.
Heb 3:14 For we have become partakers of Christ if we hold the beginning of our confidence steadfast to the end,
Heb 3:15 while it is said: "TODAY, IF YOU WILL HEAR HIS VOICE, DO NOT HARDEN YOUR HEARTS AS IN THE REBELLION."
PAUL---
1Co 10:1 Moreover, brethren, I do not want you to be unaware that all our fathers were under the cloud, all passed through the sea,
1Co 10:2 all were baptized into Moses in the cloud and in the sea,
1Co 10:3 all ate the same spiritual food,
1Co 10:4 and all drank the same spiritual drink. For they drank of that spiritual Rock that followed them, and that Rock was Christ.
1Co 10:5 But with most of them God was not well pleased, for their bodies were scattered in the wilderness.
1Co 10:6 Now these things became our examples, to the intent that we should not lust after evil things as they also lusted.
1Co 10:7 And do not become idolaters as were some of them. As it is written, "THE PEOPLE SAT DOWN TO EAT AND DRINK, AND ROSE UP TO PLAY."
1Co 10:8 Nor let us commit sexual immorality, as some of them did, and in one day twenty-three thousand fell;
1Co 10:9 nor let us tempt Christ, as some of them also tempted, and were destroyed by serpents;
1Co 10:10 nor complain, as some of them also complained, and were destroyed by the destroyer.
1Co 10:11
Now all these things happened to them as examples, and they were written for our admonition, upon whom the ends of the ages have come.
LA