Kirby, Megaman, the blue mage/mime/feral child classes from final fantasy, there is plenty to inspire the idea of a character that can copy the abilities of foes, but with the exception of an arcanist exploit and perhaps the brawler's martial flexibility, there are not really any ways to create a character purely based around that concept.
How would one go about making such a class? How do we make it balanced?
It honestly might be best to flavor it since it's really vague how the Blue Mage actually works in Final Fantasy games.
Though, I am designing a class that has a "spell thief" ability where they steal essence from a spellcaster and add a spell that caster knows to their spell list and list of spells known.
Or, like the Incantatrix class for AD&D from Dragon #90 (October 1984)?
The trick is how to balance it, I think and deciding what ability each monster or npc might grant. Like Soma Cruz or Shanoa from the Castlevania franchise, it could be quite difficult to determine what is balanced and what isn't.
If you're willing to use 3rd party, take a look at Spheres of Power's shifter class and alteration sphere. The alteration talent, Mimicry, lets you temporarily grab other alteration talents based on what you're looking at. So you absolutely could steal a lot of temporary abilities. It may not be what you're looking for, but it also wouldn't require giving every monster in the Bestiary a Castlevania style ability that you can grab.