Yes. These words concerning the chief steward over the household of the Davidic King Hezekiah are very similar to the words and ideas used by Jesus when he addressed Peter, suggesting an imperfect Old Covenant foreshadowing of a perfect New Covenant reality, namely, that Davidic King Jesus intended to make Peter the chief steward over his household, the Church:18 And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.” (Matthew 16:18-19)
Key / Keys: Isaiah 22 has “the key of the house of David”; Matthew 16 has “the keys of the kingdom of heaven.”
Open and shut / Bind and loose: Isaiah 22 has “he shall open, and none shall shut: and he shall shut, and none shall open”; Matthew 16 has “whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
Household-supporting peg (that would one day give way) / Church-supporting rock (against which the powers of death would not prevail): Isaiah 22 has “I will fasten him like a peg in a sure place, and he will become a throne of honor to his father’s house. And they will hang on him the whole weight of his father’s house, the offspring and issue, every small vessel, from the cups to all the flagons. In that day, says the Lord of hosts, the peg that was fastened in a sure place will give way; and it will be cut down and fall, and the burden that was upon it will be cut off, for the Lord has spoken.”; Matthew 16 has “And I tell you, you are Peter, on this rock I will build my church. I will build my church, and the powers of death shall not prevail against it.”
The notion that Peter was to be the chief steward over Jesus’ Church is also supported by the exchange between Peter and Jesus in Luke 12

eter said, “Lord, are you telling this parable for us or for all?” And the Lord said, “Who then is the faithful and wise steward, whom his master will set over his household, to give them their portion of food at the proper time? Blessed is that servant whom his master when he comes will find so doing. Truly, I tell you, he will set him over all his possessions. But if that servant says to himself, ‘My master is delayed in coming,’ and begins to beat the menservants and the maidservants, and to eat and drink and get drunk, the master of that servant will come on a day when he does not expect him and at an hour he does not know, and will punish him, and put him with the unfaithful. (Luke 12:41-46)And, in John 21:When they had finished breakfast, Jesus said to Simon Peter, “Simon, son of John, do you love me more than these?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Feed my lambs.” A second time he said to him, “Simon, son of John, do you love me?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Tend my sheep.” He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” And he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, “Feed my sheep. (John 21:15-17)