In answering this question, i have avoided New Testament scriptures with the exception of Jesus claim that He was God.
In Jesus time, only the old Testament existed and not in the form we know them today. Clearly, those who knew the scriptures were well aware of the relationship between an individual and His (or Her) God, as it was well established in numerous texts.
Consider the following Old Testament quotations from various individuals describing a "personal" God...
Exodus 15:2
The LORD is my strength and my defense; he has become my
salvation. He is my God, and I will praise him, my father's God, and I
will exalt him.
Daniel 9:4
I prayed to the LORD my God and confessed:
"Lord, the great and awesome God, who keeps his covenant of love with
those who love him and keep his commandments,
Psalm 18:2
The LORD is my rock, my fortress and my deliverer; my God is my rock, in whom I
take refuge, my shield and the horn of my salvation, my stronghold.
Isaiah 12:2
Surely God is my salvation; I will trust and not be afraid. The LORD,
the LORD himself, is my strength and my defense; he has become my
salvation.
Genesis 24:48
and I bowed down and worshiped the LORD. I
praised the LORD, the God of my master Abraham, who had led me on the
right road to get the granddaughter of my master's brother for his
son.
From my own Christian perspective, Jesus is "My Personal Saviour". Yes he is a man for all people so to speak, however, unlike a King or Prime Minister, Jesus door is open and we communicate directly and personally on a regular basis.
So when Thomas exclaimed, "My Lord and My God", he was stating in that moment he finally realised that the very man he had been following was indeed God! My interpretation of the Biblical narrative concerning Thomas is that prior to this moment, after Jesus had died, Thomas had lost his faith..a faith in Jesus prior claim during his ministry that he was God. That faith was now restored.
I think generally all Christians believe in an eternal God (one who also prexisted eternally). Now listen to what Jesus said concerning himself about this...
John 8:58,59
58“Truly, truly, I tell you,” Jesus declared, “before Abraham was
born, I am!”
59 At this, they picked up stones to throw at Him. But Jesus was
hidden and went out of the temple area.
There is only one reason why they (in this case Jewish Pharisees, see vs 13) would pick up rocks to stone anyone in the above event, Jesus was claiming to be "the" pre-existing eternal God, a crime punishable by stoning to death!