The inspired writer of the Hebrews' Epistle had said that Jesus, as God’s Son, had inherited a name greater than that of the angels. He pointed out that if the word of the Law that God transmitted through angels to Moses received heed, more so would the word spoken by the Son (Heb 1:1-4; 2:1-4) Then, Hebrews chapter 3 continued to stress Jesus’ superiority. We've to recall that after Miriam and Aaron complained over Moses’ major role in Israel, God called Moses ‘My servant Moses is not so, who is faithful in all mine house.’ (Num 12:7 [KJV, until further indications]; see - please - also the LXX reading)
Since Aaron as high priest had access to the tabernacle (even the 'Most Holy'), God must have been referring to the nation, of Israel as His 'house' in which Moses was His servant. The Hebrews' writer evidently alluded to this in Hebrews 3:2. He said that Jesus was faithful to God, just as Moses had been “in all his [God's] house”. That inspired writer reasoned further that “he who hath builded the house hath more honour than the house”. The epistle was saying, then, that Jesus, the Son of God, merited more honor than Moses because Jesus constructed a 'house', whereas Moses did not.
Extending the line of reasoning, the epistle said that Moses was a faithful ‘servant' [or 'attendant'] in the house (Hebrews 3:5) So, Moses was himself in or was part of the 'house' of Israel. But what of the One who would come later, the Messianic prophet whom Moses spoke about? (Deu 18:18, 19; Joh 1:21, 25; Act 3:22, 23) That prophet would be more than a faithful 'servant' in the 'house' of Israel. The inspired writer tells us that Christ was “a Son over the house of that One” that is, his heavenly Father. Hardly Jesus would be appointed over the 'house' of physical Israel. In fact, whereas Moses served in a 'house' according to the Law covenant, God promised a better arrangement under “a new covenant”. (Jer 31:31-34; Heb 8:7-13) The new covenant is with the 'house', or 'nation', of the "Israel of God" [Young], according the Paul's words in Gal 6:16. Thus, in Heb 3:6 we read: “whose house are we, if we hold fast our boldness and the glorying of our hope firm unto the end” (Hebrews 3:6 [ASV]; see - please - also 1Tim 3:15; 1Pet 2:5).
So the 'house' in which Moses served was the fleshly nation of Israel, but the 'house' that Jesus constructs and over which he serves is the ekklesia of the 'Israel of God''s members, who are maintaining their “hope firm unto the end.”