When you merely select 2 objects and align them, they will both move to match the alignment. If the align is set to "align to artboard" all selected objects move to align with the artboard. If the alignment is set to "selection", then all objects are moved to match the alignment.
In the case of align on center - The center of each object is brought into alignment. AI seems to look at the center point of all selected objects, calculates the distance between the left and right most center points, then aligns everything to the middle of that distance - causing everything to shift either right or left.
Both of these operations are shown below:

In this case, what you want is to align on a key object. Which means you can cause the top object (type) to move and align with the bottom object (box) without moving the box.
When using a Key Object you tell Illustrator you don't want the key object to move and you want things to align to the key object, not each other or the artboard.
To align on a key object:
Using the Selection Tool (Black arrow)
- Click to select the type
- Hold the Shift key
- Click the white box to add it to the current selection
- Let go of the Shift key
- Click the white box again
You should see a thicker highlight indicator on the box. This indicates it is set as the "key object". Meaning it won't move and any alignment will use its position as a base.
Then align on center... the type should move, the box should not.

The trick to using a Key Object is to click it an extra time with the Selection Tool after you have everything selected.