I have two continuous probability density functions $f_1(x)$ and $f_2(x)$ giving the probability of picking any positive real number x from those distributions. If I am going to pick one number from each distribution and multiply them, probability of multiplication to be y would be:
$$p(y)=\frac{\displaystyle \int_{-\infty}^{\infty} f_1(x)f_2\bigl( \frac{y}x \bigr)dx}{\displaystyle \int_{-\infty}^{\infty} \left(\int_{-\infty}^{\infty} f_1(x)f_2\bigl( \frac{y}x \bigr)dx \right) dy}$$
$\int_{-\infty}^{\infty} f_1(x)f_2(\frac{y}{x})dx$ resembles convolution to me, but I have never seen this operator before. I am not math major so its very likely that I wouldn't know if this is something known and I don't know how it would have been called. Forgive me if its a dumb question but is this a thing, and if so what is the name for this operator so I can study it for the project I am working on?
Your denominator should be $\int_{-\infty}^\infty f(x) dx\int_{-\infty}^\infty f(y) dy=1$, at which point your expression is correct. Intuitively, the chance of a given product is $0$ because given any $x$ chosen by $f_1$ the chance that $f_2$ yields $\frac yx$ is zero. What you be interested in is the probability density in the product, $p(y)dy$, which should integrate to $1$