Alpha to selection selects all the pixels but proportionally to their opacity (the "marching ants" are telling you where the selection level goes from
<50%). This selection is fine from many things, but not to copy/paste the contents(*). If you want to have them fully selected you have to
Select > Save to channel, threshold the saved channel with a value of 1, and then re-create a selection from the channel (right-click, and
Channel to selection). But this is not necessary, to center you contents there are at least two techniques:
Layer > Crop to content: this makes the smallest layer that contains all the non fully transparent pixels. You can then center this.
If you want a selection, use the Color selector, set
Select by: Alpha, check
Select transparent areas and set the threshold to 0. Then click on the layer's background. This will only select the fully transparent pixels.
Select > Invert and you have the selection that you originally sought.
Also, keep in mind that centering is somewhat visual when you have semi-transparent pixels on the edges. This is why in many fonts, characters with a round base (C,G,O,Q,U) slightly extend below the baseline.
(*) Because a half-opaque pixel, being half-opaque, will be half-selected, and therefore half-copied. When pasted, the half-copy of the half-opaque will be quarter-opaque which is much more transparent. This makes things look thinner.