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Specializations and "win in a fight" do not need you to talk to the slave again to advance their level as they'll level up as soon as the condition is met, but all item requests do require talking to them a second time if you did not have the item in your mansion inventory beforehand. When you already have the items for their level up before talking to them, they'll immediately take it and there's no need to talk to them again as the dialogue allows you to give them the items right there. Any time you have to actually acquire said items, like hair dye or chain mail or a longsword, you have to talk to the slave again after getting the item(s) in order to give it to them for level-up.

It's always been this way since the level-up system was changed, but I suppose it can seem confusing if you've never ran into this issue before and are still new to all the little nuances of the game and how it functions. Still, you knew you had to talk to the slave to initiate the level-up process, so you should have at least attempted to talk to the slave again after acquiring the item. Try talking to your slaves more often, you can learn a bit of how they feel about you and their situation as their loyalty, obedience, and stress changes.

And remember, items need to be in your mansion inventory, not your backpack/party inventory, in order for the game to make use of them. This goes for quests as well, not just leveling up.