Did a quick test and as far as I can see, the Increased Damage Over Time for Minions filter works fine for highlighting drops with that affix…
i.e. I crafted a weapon with the affix, created a filter that recoloured any item with that affix, then dropped my item and it is correctly recoloured as per the filter…
The affix shard name & the actual item affix do not correspond, but I would just chalk that up to the devs not updating the text… See - Search - Last Epoch Item Database and you can read that the Shard applies the correct item suffix name…
One thing I will say, is that this is rare suffix/shard drop and it requires character level 48 to use… Anecdotally, to illustrate how rare… I have been playing for a while now and I have about 95k shards stored… but ONLY 170 Damage Over Time for Minion shards… I have plenty of weapons with this affix stored tho, so it definitely does drop.
I cannot recall the exact mechanism at work here, but being close to the level of the affix also improves the chances of it dropping… if Llama or Heavy see this thread they could probably give the exact way it works re drops and character level…