P. Aluffi gives the following definition in "Algebra: Chapter $0$".
"We say that a category $C$ is a category with products if for all $A, B\in Obj(C)$ the category $C_{A, B}$ has final objects."
By $C_{A,B}$ here we mean the derived category whose objects are all triples $(X\in Obj(C),\phi:X\rightarrow A,\psi:X\rightarrow B)$. So speaking about the product $A\times B$ we mean the triple $(A\times B,\phi:A\times B\rightarrow A,\psi:A\times B\rightarrow B)$ (it also must be a final object in $C_{A,B}$). I'm interested in the following. Given a category with products, speaking about $A\times A$ can the triple $(A\times A,\phi:A\times A\rightarrow A,\psi:A\times A\rightarrow A)$ be random, or the projection morphisms $\phi$ and $\psi$ must be the same?
The projection morphisms in general are not the same.
Consider the category Set, and let $A = \{0,1\}$. Then (up to unique isomorphism) the product $A\times A$ is the Cartesian product $\{(0,0),(0,1),(1,0),(1,1)\}$. $\phi$ is the function that takes the first element of the pair, so it maps $(0,1)$ to 0 and $(1,0)$ to 1, whereas $\psi$ is the function that takes the second element, so it maps $(0,1)$ to 1 and $(1,0)$ to 0.
So they are different functions, and they have to be, in order to project out the first and second elements of the pair.
There are exceptions, such as if $A=\{0\}$ in Set, in which case $A\times A$ is $\{(0,0)\}$, and both projections are just the isomorphism that maps this back to $\{0\}$. But I think in most useful categories these exceptions will be special cases.