According to Isaiah

Arise, shine;
For your light has come!
And the glory of the LORD (YHVH) is risen upon you.
For behold, the darkness shall cover the earth,
And deep darkness the people;
But the LORD will arise over you,
And His glory will be seen upon you.
The Gentiles shall come to your light,
And kings to the brightness of your rising.
“Lift up your eyes all around, and see:
They all gather together, they come to you;
Your sons shall come from afar,
And your daughters shall be nursed at your side.
Then you shall see and become radiant,
And your heart shall swell with joy; ... “The sun shall no longer be your light by day, Nor for brightness shall the moon give light to you; But the LORD will be to you an everlasting light, And your God your glory. Your sun shall no longer go down, Nor shall your moon withdraw itself; For the LORD will be your everlasting light, And the days of your mourning shall be ended.

-- Isaiah 60:1-5

The light is "the glory of the LORD" which also affects the people so that they may also become bright.

According to John

And this is the judgment: the light has come into the world, and people loved the darkness rather than the light because their works were evil. For everyone who does wicked things hates the light and does not come to the light, lest his works should be exposed. But whoever does what is true comes to the light, so that it may be clearly seen that his works have been carried out in God.” -- John 3:19-21 (ESV)

If I (Jesus) alone bear witness about myself, my testimony is not true. There is another who bears witness about me, and I know that the testimony that he bears about me is true. You sent to John, and he has borne witness to the truth. Not that the testimony that I receive is from man, but I say these things so that you may be saved. He (John) was a burning and shining lamp, and you were willing to rejoice for a while in his light. But the testimony that I have is greater than that of John. -- John 5:31-36 (ESV)

Therefor we can conclude that John meant "the light" is the true testimony of Jesus, which reveals God's character, plan, intentions and instructions. Those who love to do evil, hate the testimony of Jesus which states that Jesus will judge them. They don't like to be corrected or admit fault. Those who receive the testimony, rejoice because the testimony states that they will be saved and enjoy eternal life.

However, later Jesus said:

Again Jesus spoke to them, saying, “I am the light of the world. Whoever follows me will not walk in darkness, but will have the light of life.” -- John 8:12 (ESV)

But even in this context Jesus means he brings the testimony of "the light", because if you continue reading:

So the Pharisees said to him, “You are bearing witness about yourself; your testimony is not true.” Jesus answered, “Even if I do bear witness about myself, my testimony is true, for I know where I came from and where I am going, but you do not know where I come from or where I am going. You judge according to the flesh; I judge no one. Yet even if I do judge, my judgment is true, for it is not I alone who judge, but I and the Father who sent me. In your Law it is written that the testimony of two people is true. I am the one who bears witness about myself, and the Father who sent me bears witness about me.” -- John 8:13-18 (ESV)

Being "the light" means to be the one who witness the testimony of Jesus. If "the light" was exclusive to Jesus only, then it would mean that John was also Jesus and that Jesus would stop being Jesus when he leaves:

"As long as I am in the world, I am the light of the world." -- John 9:5 (ESV)

More examples where Jesus took up the role of being "the light":

So Jesus said to them, “The light is among you for a little while longer. Walk while you have the light, lest darkness overtake you. The one who walks in the darkness does not know where he is going. While you have the light, believe in the light, that you may become sons of light.” When Jesus had said these things, he departed and hid himself from them. -- John 12:35-36 (ESV)

I have come into the world as light, so that whoever believes in me may not remain in darkness. -- John 12:46 (ESV)

However, the source of "the light" is God and we too can also "walk in the light" as Jesus (by witnessing as Jesus did) according to this scripture also written by John:

This is the message we have heard from him and proclaim to you, that God is light, and in him is no darkness at all. If we say we have fellowship with Him while we walk in darkness, we lie and do not practice the truth. But if we walk in the light, as he is in the light, we have fellowship with one another, and the blood of Jesus his Son cleanses us from all sin. -- 1 John 1:5-7 (ESV)

According to Isaiah

This was recorded by Isaiah many centuries before Jesus were born:

Arise, shine;
For your light has come!
And the glory of the LORD (YHVH) is risen upon you.
For behold, the darkness shall cover the earth,
And deep darkness the people;
But the LORD will arise over you,
And His glory will be seen upon you.
The Gentiles shall come to your light,
And kings to the brightness of your rising.
“Lift up your eyes all around, and see:
They all gather together, they come to you;
Your sons shall come from afar,
And your daughters shall be nursed at your side.
Then you shall see and become radiant,
And your heart shall swell with joy; ... “The sun shall no longer be your light by day, Nor for brightness shall the moon give light to you; But the LORD will be to you an everlasting light, And your God your glory. Your sun shall no longer go down, Nor shall your moon withdraw itself; For the LORD will be your everlasting light, And the days of your mourning shall be ended.

-- Isaiah 60:1-5,19-20

Even here, we can see the light is not exclusive to God only. God's "glory will be seen upon you" and this will attract the "Gentiles".

According to Matthew

Jesus said:

"You are the light of the world. A city set on a hill cannot be hidden. Nor do people light a lamp and put it under a basket, but on a stand, and it gives light to all in the house. In the same way, let your light shine before others, so that they may see your good works and give glory to your Father who is in heaven." -- Matthew 5:14-16 (ESV)

According to Paul

For so the Lord has commanded us, saying,

“ ‘I have made you a light for the Gentiles, that you may bring salvation to the ends of the earth.’ ”

And when the Gentiles heard this, they began rejoicing and glorifying the word of the Lord, and as many as were appointed to eternal life believed. And the word of the Lord was spreading throughout the whole region.

-- Acts 13:47-49 (ESV)

and

If you are sure that you yourself are a guide to the blind, a light to those who are in darkness, an instructor of the foolish, a teacher of children, having in the law the embodiment of knowledge and truth — you then who teach others, do you not teach yourself? -- Romans 2:19-21 (ESV)