I don't know if I'm remembering correctly, but is there a monster or monsters in FFVI that always runs away when you try to fight it?
If yes, what is it (they) called? Thanks!

Welcome!

Although I shamefully haven't played FFVI in a long time, I do remember certain enemies do tend to escape from battle - particularly human enemies like Imperial soldiers (this is particularly true early in the game in South Figaro where Locke is trying to find a disguise - after stealing an opponent's clothes, the enemy turns into a different enemy labelled simply 'Birthday Suit', who will then escape a few turns later.

Certain other human enemies like mages and thieves will also run away when other enemies in the party are defeated and you don't take them out quickly enough. Cactuars also escape from battle after a few turns too if I remember right.

Some of the guard dogs usually run away when you kill the guardsman.

Doesn't cactuar, though I've forgotten where you fight them (they are in world of balance)?

Herman Munster?

Yeah the guard dogs. IIRC I went back to Narshe way later in the game LV40+ and they flee.

Yes there is an enemy in the world of ruin that has spells to learn from control.They look like squirrels a little and run away enforce you can often get anything from them.