How did they come to meet ?

Answer

is

It was Krauncha, the Gandharva - celestial musician, who ends up as Ganesha s mouse. 

https://m.freepressjournal.in/article/cmcm/why-mouse-is-the-vahana-of-lord-ganesha/c6590fce-91a9-4727-8afa-b00d9a583027

I came across another story, where as a kid Vinayagar, pesters his dad for a pet /vahana as his brother already had one. Shiva catches a mouse that keeps disturbing him, and hands it over to Vinayagar. 

As for other religions, Apollo of the Greeks, was also associated with mice.

 A new religion, in fact a religious parody, has taken inspiration from a Mouse. Check out Dinkoism, if you are interested.