For all the naysayers: Pretty much any normal attack the Skeletons do will be considered "with this Weapon". The weird wording is likely to avoid silly edge cases such as Minion Instability gaining Triple Damage.
As such, all Added Damage effects will be Tripled here (3x). With Unholy Might, anything that gives Added Physical Damage is instead effectively 3.9x stronger. That includes Carrion Golem, Abyss Jewels, Rallying Cry and somewhat notably Champion's passive skill Master of Metal.
Converting all Physical Damage to Chaos will also enable some shenanigans with the built-in Wither debuff. An army of 12 Skeletons with just the 21% chance on the sword will, on average, maintain 7 Wither stacks (or 42% more damage, effectively). Add in a Withering Touch gem to your link and you're looking at all 15 stacks for a whopping 90% More damage; now 1 point of Added Physical Damage translates to 7.41 Chaos damage before even factoring in support gems.
With proper gear support this thing is gonna make for fun builds. It could even be a nice way to play a Champion summoner that's not totally trash.