Aggro System

I honestly think so at times when running some nodes, but don't take my word for it. Just a theory since I run my Tamamo lancer in a node I thought was filled with only Archer enemies and has a low chance of saber ones, but sabers appeared 2 out of 3 mobs and was forced not to use her cards, but a support waver and my Euryale's cards, but the time I brave chained with Tamamo lancer they started hitting her.
Well with that said I think it just increases the chance and further testing lead me to an answer of no since sometimes instead of attacking my Drake they focused on a support waver again don't take my words for this, but I think it's more of RNG and if they did have a system like that they would've told the players about it though I honestly feel like there is.
Sometimes I felt they even attack and focus on my party members they know they had an advantage against which made me tired using zerkers.

I know some fights like Lancelot in Camelot would target the highest HP. I've seen some fights where they attack the highest NP gauge. But I think specific aggro systems are only for event and main quests. Free quest are random as far as I can tell.

This may just be in my head but I do think some servants will target others specifically. For example it seems like when ever I fight Mordred or Zerkerlot they prioritize Altria.

It does seem to apply only in the occasional story boss fights, though it can’t be confirmed. I have seen no case of it in normal battles though. Far to many times I have had servants one hit from death and they are not targeted for several turns.

I even heard that Merlin is always targeted by enemies because he is too broken...

You'll usually only find an aggro mechanic in nodes that require you to defeat other servants. Majority of the time, you'll end up running into these when it's story based but Saber Wars has, by far, the most enemies that have target focus based on your comp. Feel free to add on if I missed any.

Servants:
Carmilla focuses Females
Lalter focuses Mordred
MHX specifically focuses sabers
Brynhildr focuses servants with "Brynhildr's beloved" trait
Nobu has an affinity towards attacking "divine" trait servants
Mordred focuses Altria (& servants with the 'Arthurian' trait)
Lancelot focuses Altria
Boudica focuses servants with the "Roman" trait

Non-Servants:
Vengeful Spirits have affinity towards the last servant to attack
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As an endnote, focus doesn't necessarily mean 100% of the time. Servants that i've listed are just more likely to be attacked by an opposing servant based on their traits. This is gathered from my experience with 2 campaign playthroughs and some events. That's all I got.