f(x)f(y) − f(xy) = x + y

Substitution $y=0$. $f(x)f(0)-f(0)=x$, i.e.
$$f(0)\bigl(f(x)-1\bigr)=x\quad\text{for all }x.$$
If $f(0)=0$ the left side is $0$ for all $x$, which is absurd (take $x=1$). So $f(0)\ne0$ and
$$f(x)=1+\frac x{f(0)}.$$

Determining the constant. Setting $x=0$ in this formula gives $f(0)=1$. Hence $f(x)=x+1$.

Answer. $f(x)=x+1$ is the only solution.

Lesson. Always: (1) try $x=0$, $y=0$, $x=y$, $y=1$; (2) when an equation is solved by a formula, plug the formula back to check it is a solution.