How far can Shakespeare's The Tempest be seen as a colonization?
I think the answer to this is that you can see it as colonization as far as you want. Many people have looked at it in this way.
To these people, Prospero is playing the role of the Europeans. He has come to this "new" land and has taken it over. He has brought "civilization" to the land because he is using his knowledge (the power in his book) to control the spirits who are native to the island. This can be seen as a metaphor for the use of European knowledge to conquer the New World.
We can also look at Caliban as a figure in the racist imaginings of Europeans. He is seen as a native of the island -- like the Native Americans perhaps. He is described as being ugly (barely even human) and he is enslaved.
Overall, then, you can say that this play portrays a "superior" European who uses his knowledge to conquer and enslave the "subhuman" beings he finds in a new world or new land.
I would have to agree with the previous answer as far as it goes.
When Prospero makes his decision NOT to get his vengeance against his brother and the King, his whole attitude changes. He forgives them. He breaks his staff of power and throws his book into the sea. He forgives Caliban and bequeaths the island back to him.
All this implies that Caliban has learned and once the foreigners leave, the island would be his. With no foreigners, no colonization.
If Caliban, a native of the island, is compared to the Native Americans, what of Ariel? He, too, is native to the island and is a magical character and usually visualized as quite pleasing looking (except when he is the Harpie, of course).
At the end of the play everybody except Caliban returns home abandoning the "new world".