Why do people want Blue Mage in this game?

[VanilleFang]

Strago [Is that how his name was spelled? Relm's father] (FF6) and Gau (FF4) were also BLUs in a sense though not by name (they both learned abilities from monsters)

Beastmaster summoned monsters from pots in FFXI. They were in other FFs too, V had one IIRC

Strago is Relm's grandfather not father, and Gau is from FFVI as well. There is no blue magic in FFIV, except for "Jive" in After Stories.

To clarify the following games/characters have BLUE MAGIC (or some varient).

FFIV After Story: Calca has "Jive" (which is random).
FFV: All characers can be a Blue Mage, and may have "Blue Magic" set when using another job as well.
FFVI: Strago has "Lore". Gau's "Rage" is completely different than Blue Magic, as it replaces "Attack" and you lose control of him. Gau never learns any attack he uses.
FFVII: Everyone can use the "Enemy Skill" materia.
FFVIII: Quistis has "Blue Magic" as her Limit Break.
FFIX: Quina has "Blue Magic".
FFX: Kimahri has "Ronso Rage" as his Overdrive.
FFX-2: The Gun Mage dressphere has "Blue Bullet".
FFXI: Blue Mages have "Blue Magic".

There's also a few other, none main series games as well.

I'd also like to point out that the OP dumbed down how Blue Magic is learned. In some games you do learn it by taking a direct hit by the spell/ability, but in other games it's different. In FFVI Strago just has to be in the battle when the abilities is used, in FFVIII you have to use items on Quistis, in FFIX Quina eats enemies, and in FFX Kimahri has to use Lancet.