Each modern Fromsoft game utilizes game engines that have evolved in a nearly straight line from Demon's Souls to Armored Core 6 (with one major hiccup featuring Dark Souls 2), so while I have only directly tested this in DS1, game data, typical carryover behavior of these systems, and accounts of the behavior in other games heavily imply it's the same (or at least, nearly identical) in every Fromsoft game (sans DS2).
With that out of the way, here's the details on TargetPriority:
TargetPriority essentially adds a % chance at some mystery interval (that is fairly frequent) to target the character with the TargetPriority modifier instead of whoever they are currently targeting.
Shabriri's Woe has a value of 0.1, which means a 10% chance to make an enemy switch their target from somebody else to whoever has Shabriri's Woe equipped.
Some conventional examples of TargetPriority in ER:
Shabriri's Woe: +10%
Howl of Shabriri: +10%
Cursed-Blood pot: +1000% (effectively 100%)
Duelist Helm: +3%
Duelist Greaves: +3%
Duelist Cloak: +4%
(Rotten Duelist set is identical to Duelist set)
TargetPriority is also capable of basically the exact opposite effect, and add a chance to make characters switch targets away from somebody. Interestingly, Spirit Summons have this behavior. Some summons have a 50% chance to make enemies switch targets away from them, others have a 25% chance.
Normal enemies in Elden Ring also utilize TargetPriority. Dragons, for instance, are basically guaranteed to be targeted by NPCs vs any other nearby enemy. Giant land squirts are very, very likely to be ignored vs any other nearby enemy. There are seemingly other cases as well, but I haven't tracked them down at this time.
Now for some examples in earlier Fromsoft games.
Demon's Soul's Ring of the Accursed seemingly behaves in EXACTLY the same way as Shabriri's Woe, with +10%. Neat.
Dark Souls 1 has several interesting cases.
Moonlight Butterfly is basically guaranteed to target the host over summons.
Young Sif in the DLC has a value of -70%, making Manus 70% more likely to stop targeting Young Sif and target the player (or another player summon) instead.
Being directly hit by an Alluring Skull adds +10% to that enemy, making other NPCs slightly more likely to target that one enemy. This is QUITE an obscure condition and effect.