Let $\otimes: C\times C\to C$ be the bifunctor of a monoidal functor $C$ with left unitor $\rho$ and right unitor $\lambda$, and identity object $e$.
By using the naturality of $\rho$ I can show that if $f_1\neq f_2$, then $1_{e}\otimes f_1\neq 1_e \otimes f_2$, hence $g\otimes f_1\neq g\otimes f_2$ for any morphism $g$. Similarly if $f_1\neq f_2$, then $f_1\otimes g\neq f_2\otimes g$.
However is it true if $f_1\neq f_2$ and $g_1\neq g_2$, then $f_1\otimes g_1\neq f_2\otimes g_2$?
There are counterexamples even to the cases you claim to have solved. For instance, if $C=\mathtt{Set}$ and $\otimes$ is the categorical product then all functions become the same when you take the product with a function whose domain is the empty set.
Or, for a counterexample where both pairs of maps are distinct, if $C=\mathtt{Ab}$ and $\otimes$ is the tensor product, you could take $f_1,f_2:\mathbb{Z}/(2)\to \mathbb{Z}/(2)$ to be the two different maps and $g_1,g_2:\mathbb{Z}/(3)\to\mathbb{Z}/(3)$ to be the two different maps and then $f_1\otimes g_1=f_2\otimes g_2$ since $\mathbb{Z}/(2)\otimes\mathbb{Z}/(3)$ is trivial.