I was reviewing this question and got motivated to solve this general problem:
Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y)=\alpha\big(f(xy)-f(x)f(y)\big)$$ where $\alpha$ is a nonzero real constant.
I found out that it can be solved in a similar way to what I did in my own answer to that question. It was interesting for me that we don't need any regularity conditions like continuity. As I didn't find any questions about the same functional equation on the site, I thought it might be useful to post my own answer to it.
I appreciate any other ways to solve the problem or any insights helping me understand why there's no need of regularities while some functional equations like Cauchy's need such additional assumptions to have regular solutions.