"For I also am a man placed under authority, with soldiers under myself..." v8
The emphasis is on the word 'also'. This is what links the two verses/phrases - both the centurion and Jesus were 'under' authority. They had this in common. Jesus seized on the centurion's realisation of this truth to make a point about faith and this reality they shared.
When one understands authority - not the man-made kind the Pharisees ruled with, but the divine mandate that Jesus operated under, then authority is good. It is for our good and not our harm or disadvantage.
We can easily think bad of authority as it is 'lorded' over us - not for our benefit but for sake of those wielding the power. Sadly this is what mankind has become accustomed to.
Jesus wielded authority too, but it was by love, respect and justice because he represented God when he healed, made wine, fed thousands etc.
The centurion was aware of the power Jesus exercised. He was able to grasp that Jesus was also under authority as he was. He saw the fruits of Jesus life and the events that occurred by his hand and word and noticed there was a power for good working through him.
"I say to you, not even in Israel have I found such great faith.” v9
The centurion's understanding was regarded as faith by Jesus - marvelling that not even amongst God's people was this understanding and trust/faith evident.
Notice Jesus 'turned' and said this to those Jews/Israel who followed him - a reprimand of sorts for their 'little' faith considering what they should have had with their entire backstory, experiences and heritage as God's people of promise.
I praise you, Father, Lord of heaven and earth, because you have hidden these things from the wise and learned, and revealed them to little children. Matt 11:25
Throughout Jesus' short ministry, God allowed several 'gentiles' to understand and leap past the Jews' hindered view of what was going on in their midst - the Messiah that they waited for was here, but they saw and believed him not.
When Jesus spoke, his word carried the weight of the One over him - his Father and God. The centurion probably never saw the depth of this reality, but he did understand the concept and that was enough to believe in. It was enough to have his servant healed.
Jesus, by his response, affirmed that he indeed was under the authority of God.
But when the crowds saw this, they were awestruck, and they glorified God, who had given such authority to men. Matt 9:8
Christ never took glory for himself - he always directed glory to God - to God's power, God as the miracle worker etc. We see further on this expressed.
A sense of awe swept over all of them, and they glorified God. “A great prophet has appeared among us!” they said. “God has visited His people!” v16
This latter phrase is not of course to assume that Jesus is somehow God, but that he represents God - just as a (Godly) prophet always did.
Jesus' response points to the matter of him being under the authority of God and of the faith that is dependant on that reality - not of God directly doing good amongst them, but of God through a man, Jesus. Jesus persistently points to God in all things. Jesus' faith, trust, dependence, reliance is exactly what he wants us to understand and live by because it is only through him that we might come to the Father.
Jesus said to him, “I am the way, and the truth, and the life; no one comes to the Father except through Me. John 14:26