I'm going to be the black sheep here and a bit of a "me too" poster, but with quite a bit more to it. I really hate "me too" posts as they contribute nothing except show that it's an issue (which is what making the thread is for). So here's a bunch of things that I did/think! Woo!
First off, yes, me too but my issue is in a different modpack. Before we go and say that I don't belong or that that is the reason why, etc., I might be able to give more insight into this. I've looked around online myself and it would seem the creator intended for items (or at least armor) to be repairable through the same tier of essence that was used to make it, as in AngelicDemons's case Imperio for Imperio and whatnot. I think I recall him saying there was a bug and he would fix it, but 1) I don't know what version that was for and 2), I don't know what version that either AngelicDemons or I are using.
Using this information, I've tested it in many different ways through all sorts of anvils, to no avail. I checked the config files to see if there is a disable repair option and there is not. I've used the raw essence and the ingot. I've looked all around and the general consensus (though it's extremely hard to find this issue extensively) is to just use the same tier material/essence as previously stated.
So my thinking is that if there is a recipe for repairing like there is for crafting, perhaps that is disabled in the modpack? Attempting to look it up creates a headache for me as all I get are the general "How to repair with an anvil" garbage. Most likely my search terms, but when you put "anvil" and "recipe" along with "minecraft", you'll just get a recipe for how to make one. Put "repair" and "recipe", you get the "how to repair" junk. I'm not a modmaker, but I know that it's possible to edit how these work in a mod somehow. I've seen things disabled before so there has to be something that's getting in the way.
As it stands, my theory is that there's an anvil repair recipe somewhere that is missing, has repair disabled entirely, partially for either tools or armor, or may have a typo in it somewhere that makes it so that lines of code are ignored by accident. Maybe I'm wrong, but I hope it sparks some ideas.
Since new members don't appear to have the ability to edit per AngelicDemons, I'm gonna pray that this is all I have for now because I'd much rather not double post if I can avoid it.
Edit: Well I seem to be able to make an edit just as a test, but the edit timer might expire after a set time instead; some forums have that feature instead. Don't know, but I'm happy to see that it does exist as an option, at least for the short term until further confirmed.
As an addition, my other thoughts regarding conflicting mods is an extremely unlikely scenario as the repair requires a tool that exists only in an addon made by the core modmaker. There would only be three locations of conflict as far as I can see: 1) The addon mod not being attached to the Anvil properly (again, not a modmaker so don't ask me how), 2) The core mod and the addon having some sort of conflicting ID's in it's anvil repair "recipe", or 3) If any other mods change the anvil and how it works, then that might cause an issue since the code may have been written for the vanilla anvil instead. I can't see any mods outside of the cor having any correlation to the addon so that's ruled out. If none of the modpacks make changes to the anvil, perhaps it's a typo somewhere in the hook for the anvil regarding the core and/or the addon (typos are killers in coding. I took Java and going through a friend's code to fix it, missing ONE CAPITAL broke the entire thing), or mismatched item ID's or additional/missing letters in recipes/filenames/whatever. Again not pretending to know how everything works, only giving what I think is going on so it might help someone else.
Way later edit: I've finally gotten to the point of having Zivicio armor, and can confirm that using the essence, not ingot, repairs the armor. That being said, there must be an issue regarding the TOOL vs the anvil, since the armor repairs. Either that or perhaps it was made so tools weren't repairable as Zivicio tools are indestructable, and the intent was to have you get to that tier tool anyway. Just another thought as I got to where I am.