Is it possible to align the content inside a slice? How?
No — and yes.
slice isn't essentially a grouping element, so you can't select a slice and tell it to align its contents. Consider slices as a grid (or as a wireframe) on top of your design, that doesn't know anything what's underneath. A slice could e.g. slice a text element so that its bound could go straight through a character if you want it to go — the slice doesn't care.
However, you could
group your layers so that each
group holds the contents of a slice. Unfortunately, there's no automated process for this out–of–the–box.
Grouping by slices may not be straightforward if you already have implemented a grouping scheme for your design, but if you really need alignment per slice basis, you might want to reconsider your scheme. An alternative to grouping could be converting the layers as a
smart object and then aligning its contents.