Does Jesus have two kingdoms?

The answer to your question appears to lie in a single verse of Scripture:

All Scripture is quoted from the King James translation, unless otherwise noted.

John 18:36 Jesus answered, My kingdom is not of this world: if my kingdom were of this world, then would my servants fight, that I should not be delivered to the Jews: but now is my kingdom not from hence.

Jesus was clear in his answer to Pilate that his is a Spiritual Kingdom and not an Earthly Kingdom. My kingdom is not of this world: conversely the Kingdom of David does refer to an Earthly Kingdom. David's kingdom was of the Jewish Nation, but the kingdom of Heaven Jesus described in:

Matthew 18:23 through 35 Therefore is the kingdom of heaven likened unto a certain king, which would take account of his servants. 24 And when he had begun to reckon, one was brought unto him, which owed him ten thousand talents. 25 But forasmuch as he had not to pay, his lord commanded him to be sold, and his wife, and children, and all that he had, and payment to be made. 26 The servant therefore fell down, and worshipped him, saying, Lord, have patience with me, and I will pay thee all. 27 Then the lord of that servant was moved with compassion, and loosed him, and forgave him the debt. 28 But the same servant went out, and found one of his fellowservants, which owed him an hundred pence: and he laid hands on him, and took him by the throat, saying, Pay me that thou owest. 29 And his fellowservant fell down at his feet, and besought him, saying, Have patience with me, and I will pay thee all. 30 And he would not: but went and cast him into prison, till he should pay the debt. 31 So when his fellowservants saw what was done, they were very sorry, and came and told unto their lord all that was done. Mat 18:32 Then his lord, after that he had called him, said unto him, O thou wicked servant, I forgave thee all that debt, because thou desiredst me: 33 Shouldest not thou also have had compassion on thy fellowservant, even as I had pity on thee? 34 And his lord was wroth, and delivered him to the tormentors, till he should pay all that was due unto him. 35 So likewise shall my heavenly Father do also unto you, if ye from your hearts forgive not every one his brother their trespasses.

This is descriptive of a Spiritual Kingdom, where God is the unopposed ruler, and judges in truth and justice. That justice is so equal that it is proportional to our sense of justice.

The following is a quote from the whole Bible commentary by David Guzik:

2. (20-24) The Lord lifts up Eliakim instead of Shebna. "Then it shall be in that day, that I will call My servant Eliakim the son of Hilkiah; I will clothe him with your robe and strengthen him with your belt; I will commit your responsibility into his hand. He shall be a father to the inhabitants of Jerusalem and to the house of Judah. The key of the house of David I will lay on his shoulder; so he shall open, and no one shall shut; and he shall shut, and no one shall open. I will fasten him as a peg in a secure place, and he will become a glorious throne to his father's house. They will hang on him all the glory of his father's house, the offspring and the posterity, all vessels of small quantity, from the cups to all the pitchers."

a. Eliakim the son of Hilkiah is mentioned in passages like 1Ki. 18:18 and Isa. 36:3 as another assistant to King Hezekiah. He should be distinguished from Eliakim the son of Josiah, who was a puppet king established by Pharaoh (2Ki. 23:34).

b. My servant: What a glorious title for Eliakim! Both Shebna and Eliakim were servants of Hezekiah, but Shebna's heart was directed towards selfish ambition and glory, and Eliakim's heart was turned towards the Lord.

c. The place of Eliakim the son of Hilkiah before Hezekiah is somewhat obscure in the Scriptures; he is only mentioned in six passages, and the only description of him is that he was over the household (1Ki. 18:18, 37 and Isa. 36:3, 22). But Eliakim was famous in heaven! He shall be a father to the inhabitants of Jerusalem and to the house of Judah.

d. The Lord would take the office and authority of the unfaithful Shebna, and give it to Eliakim instead (I will clothe him with your robe and strengthen him with your belt; I will commit your responsibility into his hand). God will get His work done! If a Shebna is unfaithful, the Lord will remove him from his office, strip him of his authority, and give it to another.

e. Because Eliakim is the Lord's servant, the Lord will give him great authority: The key of the house of David I will lay on his shoulder. In that day, the chief royal steward would have the large master key of the palace fastened to the shoulder of his tunic. The key was a picture and demonstration of the authority of the chief steward. Here, the Lord gives Eliakim the authority to open and shut as the Lord's representative, which no man can oppose.

i. In this, Eliakim becomes a prophecy of the Messiah, because Jesus told us this passage spoke of Himself: These things says He who is holy, He who is true, "He who has the key of David, He who opens and no one shuts, and shuts and no one opens." (Rev. 3:7) Jesus is the one with the keys of Hades and of Death (Rev. 1:8), who has all authority both in heaven and on earth. Jesus delegates this authority as it pleases Him (Mat. 16:19).

As is stated above Jesus delegates his authority:

Matthew 16:19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

Not only to Peter alone as a man, but through his foreknowledge to the Church which Peter will later form. It is to be noted that all current Denominations are an off spring of the that first Roman Church now known as the Roman Catholic Church. Although many will argue that the church in Rome is an offspring off the church in Jerusalem, the Roman Church is most responsible for the spread of the Gospel throughout the World. It was the Roman church which influenced Constantine to declare Christianity as the official religion of Rome which began the spread of Christianity throughout the Roman Empire.

Hope this helps.

I am assuming you think that there are two kingdoms because "keys" is plural. But plurality of keys doesn't imply a plurality of kingdoms. The keys can be interpreted, for instance, as preconditions for entering the kingdom. Tertullian (circa 208) interpreted the keys as repentance and baptism, saying Peter turned them in Acts 2:38, binding the sins of all who would not submit to those and loosing the sins of all who would.

From his treatise On Modesty, Chapter 21 (translation from the Ante-Nicene Fathers series edited by Philip Schaff):

If, because the Lord has said to Peter, “Upon this rock will I build My Church,” [and] “to thee have I given the keys of the heavenly kingdom;” or, “Whatsoever thou shalt have bound or loosed in earth, shall be bound or loosed in the heavens,” you therefore presume that the power of binding and loosing has derived to you, that is, to every Church akin to Peter, what sort of man are you, subverting and wholly changing the manifest intention of the Lord, conferring (as that intention did) this (gift) personally upon Peter? “On thee,” He says, “will I build My Church;” and, “I will give to thee the keys,” not to the Church; and, “Whatsoever thou shalt have loosed or bound,” not what they shall have loosed or bound. For so withal the result teaches. In (Peter) himself the Church was reared; that is, through (Peter) himself; (Peter) himself essayed the key; you see what (key): “Men of Israel, let what I say sink into your ears: Jesus the Nazarene, a man destined by God for you,” and so forth. (Peter) himself, therefore, was the first to unbar, in Christ’s baptism, the entrance to the heavenly kingdom, in which (kingdom) are “loosed” the sins that were beforetime “bound;” and those which have not been “loosed” are “bound,” in accordance with true salvation; and Ananias he “bound” with the bond of death, and the weak in his feet he “absolved” from his defect of health.

That's not his entire interpretation as can be seen from the mention of Ananias, but certainly the idea here is that Peter was given a plurality of keys but to only one kingdom.

Short answer: no. He does not have two kingdoms currently.

John 18:36

"Jesus answered, My kingdom is not of this world: if my kingdom were of this world, then would my servants fight, that I should not be delivered to the Jews: but now is my kingdom not from hence." (KJV)

Jesus' kingdom is, for right now, not on earth. His kingdom resides in the heavenly realms and in the hearts of believers. You cannot grab or feel His kingdom right now.

However, there is coming a time after Satan is cast into the lake of fire that Jesus will have a graspable kingdom.

Rev 21:10

"And he carried me away in the spirit to a great and high mountain, and shewed me that great city, the holy Jerusalem, descending out of heaven from God" (KJV)

The Bible goes on after this verse to describe the city with the author's sight, such as sparkling, and the author goes on to give dimensions. This is clearly a kingdom able to be seen and felt. A graspable kingdom.

So the long answer is: Jesus has two different kingdoms, but only one right now.