I received Azura's Star and went after Umbra so as to be able to recharge items myself. I trapped a Dagoth something in the Azura's Star and tried to use it to recharge a weapon. However the enchant points on the weapon did not go up, they stayed the same like it had no effect. I tried reloading several times, sometimes the enchantment fails and so nothing happens (as it should), but sometimes it does work (different sound effect plus I go up a level in enchant when I succeed, since my enchant level is pretty low, around 15) however the item is not recharged.

Is this a bug?

It might be, but it's not something that I've ever encountered. I can suggest a couple of things to check, though.

Go find yourself a couple of Petty Soul Gems - they're common loot drops in crates. Go Soul Trap a Mudcrab or Nix-Hound (something small). When you've got a couple of them, burn off a couple of charges from a standard item like a Ring of Fireballs or something. Then open your inventory and drop one of the cheap-o soul gems onto your character. Select the "Recharge" button and then select the item. Assuming that you've got a standard item and it's at least partially depleted, you should see the number of charges go up by one or two. If it does, then the culprit is most likely going to be Azura's Star or the item you're trying to recharge or both. In that case, I'd take a look at any mods you might be running.

Sorry it's not a solution, but hopefully it will help you identify the problem and you can work from there.

if you have a low enchant score, this seems to regularly happen. Not sure if it is a bug, but when you get higher enchant score you'll see greater increase of charge.
It is a good idea is to grab up on cheap soul gems (you can buy replenishing soulgems at enchanter in mournhold), and soultrap weak enemies; and use them simply for raising your enchant score.

ST