Bonuses from skills don't seem to show up in Sector Space or on the ground -- only when you're actually in control of the ship in a social or mission map. Try checking in Sol System (or around some other station) or a space mission map.
Also, if you check under your Defense Stats on the ship status screen, you can see your hull and shield HP. That should show you what you have, if the mouseover on the equipment itself doesn't show the difference.
That's odd. Confirm you were checking it on a (non-Sector) space map, and not a ground map? The effects of my Structural Integrity skill are showing up just fine in space (i.e. Sol System map), but don't show up on the ground (viewed on the ship Status page) or in Sector space. That's been true even before the skill revamp.
If it's a bug, it's been there since well before Season 4 launched. I think the issue is that skills are only applied on maps where they make a difference. I haven't checked, but I bet the Captain's ground stats aren't modified on space maps either.