Why the doubling of the 1st person pronoun? A single one would have be sufficient.
Obviously, 2 subjects are addressed.
Your conclusion doesn't follow. Thomas is addressing Jesus only. That he repeated the first person pronoun is for emphasis, not because he is addressing two subjects. There is only one subject, Jesus.
There is also more context that you are not taking into account, even though Thomas's statement does stand on its own. Look at what Thomas first says when 'the other disciples told him, "We have seen the Lord."'
Joh 20:24 Now Thomas, one of the twelve, called the Twin, was not with them when Jesus came.
Joh 20:25 So the other disciples told him, “We have seen the Lord.” But he said to them, “Unless I see in his hands the mark of the nails, and place my finger into the mark of the nails, and place my hand into his side, I will never believe.” (ESV)
Notice Thomas's demands in verse 25:
1. Unless I see in his hands the mark of the nails,
2. and place my finger into the mark of the nails,
3. and place my hand into his side,
4. I will never believe.
Then:
Joh 20:26 Eight days later, his disciples were inside again, and Thomas was with them. Although the doors were locked, Jesus came and stood among them and said, “Peace be with you.”
Joh 20:27 Then he said to Thomas, “Put your finger here, and see my hands; and put out your hand, and place it in my side. Do not disbelieve, but believe.” (ESV)
Notice Jesus's response in verse 27:
1. Put your fingers here,
2. see my hands,
3. put out your hand, and place it in my side.
4. Do not disbelieve, but believe.
Jesus directly responds to each of Thomas's demands with commands to do exactly what Thomas stated. This shows the omniscience of Jesus, to which Thomas responds:
Joh 20:28 And Thomas answered and said to Him, My Lord and my God! (LITV)
Note that Thomas "said to Him;" that is what appears in the Greek. He was directly addressing Jesus. More than that though, is Jesus's response:
Joh 20:29 Jesus said to him, “Have you believed because you have seen me? Blessed are those who have not seen and yet have believed.” (ESV)
As I have stated already, Jesus doesn't rebuke Thomas. Instead, Jesus says, "Have you believed because you have seen me?". Believed what? First, that he has risen from the dead, and second, that he is God. It was Thomas's statement--"My Lord and my God"--that was the evidence that Thomas now believed, having come out of his previous unbelief. A lack of rebuke from Jesus for Thomas calling him his God, is proof that Jesus is God.
And, again, this fits perfectly with John 1:1. John beings his gospel by stating that the eternal Word is God, who then takes on human flesh, and ends his gospel with Thomas claiming that the incarnate Word, Jesus, was his God. That is a consistent message throughout John's gospel.