Is projection a quotient function?

Question: Is projection a quotient function?

Answer: No.

Why? Why would it be?

If $p: X×Y → X$ is the projection, then there is no equivalence relation in the first place to talk about quotient function.

A better question: What equivalence relation on $X×Y$ is created by the projection $p: X×Y → X$?

Answer: $(x,y) \sim (x,y')$.

In this case the projection is the quotient function of this quotient set: $$p(x,y) = [(x,y)] = x.$$

You can ask this question about quotient maps but it has nothing to do with topology. You can ask this question about quotient linear operators but it has nothing to do with linear algebra. You can ask this question about quotient homomorphisms but it has nothing to do with group theory.