Are there alternatives functions for the gamma function that was used as generalisation for the factorials?

Question

Now asked on HSM here.

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$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ \ \ x>0. $$

Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties

1. $f(x+1)=xf(x)$
2. $f(1)=1$
3. $f(x)$ is a continuous
4. $f(x)$ is log convex

But before Bohr and Mollerup why did Euler or Bernoulli chose this function ? there are infinite many positive continuous function $f$ st $f(x+1)=xf(x)$, was there any alternatives functions for the gamma function as a generalisation for the factorial function that was used before and now we stopped? If there is one then why do we stop using it? If there isn't one then why and how did Euler and other found the "Only" "natural" generalisation for the factorial function centuries before Bohr and Mollerup as somehow they knew that this solution is unique.