Q:

Can we change the text align without moving the position of an object in illustrator?

Is there such an option we can turn on so when we change the text align of an text object the position of this object will not move in Illustrator?

A:

You can realign the point text with the original. This could be useful if you ever used point text an need a second line below aligned. Do this:

select your text.
hit copy.
realign your text (yes it moves).
paste in front.
align the realigned text with the align tool.
delete the old copy.

Animation 1: This is what the described sequence gets you.

A:

When we change the text align of an text object of "Point Type" the position of this object will move. In order to avoid this behavior, we have to change the type of the text block to "Area type". Go to "Type" menu and select "Convert to Area Type". 

