Uniqueness of projection map of categorical product

No.

Let $X=Y=P=\mathbb N$. $\mathbb N$ can serve as $\mathbb N\times\mathbb N$ in many ways. Pairing function describes one, but it makes it clear that there are others, some of which you can see in the references.

The answer is no.

If $(P,\pi_1:P\to X,\pi_2:P\to Y)$ is a product, and $\phi:Q\to P$ is an isomorphism, then $(Q,\pi_1\circ\phi,\pi_2\circ\phi)$ is also a product (this isn't hard to check). In particular, if $Q=P$, this means any automorphism on $P$ gives you a new pair of projections, with respect to which it is a product of exactly the same objects.

The trick is that the unique pairing maps will change with respect to the different choices of projection. Given $f:A\to X$ and $g:A\to Y$, let's write the unique map to $(P,\pi_1,\pi_2)$ as $\langle f,g\rangle$; but for the product obtained via composing the projections with an automorphism $\phi$, the appropriate pairing map is now $\phi^{-1}\circ\langle f,g\rangle$.