Comment by xanderlewis

I agree it’s not really used symbolically, but isn’t it referring to the difference between, say:

(x + y)(x - y) = x^2 - y^2

(an identity, since it’s true for every x and every y) and something more arbitrary like

3x^2 + 2x - 7 = 0

(an equation certainly not valid for all x and whose solutions are sought).

Of course, really, the first one is a straightforward equality missing some universal quantification at the front… so maybe that’s just what the triple equals sign would be short for in this case.