Question says it all. The reason I asked this is because I got a shiny on my 3rd stage in the game, and that made me wonder has the rate been officially confirmed?
Unfortunately, the odds have never been confirmed or officially commented on.
As of August 2021, there's only been assumptions, but nothing offical.
Source: Lot's of looking into this
The odds are speculated at being 1/8192 but they've never been confirmed. Shinies in Rumble aren't uncommon, however, trying to get SPECIFIC shinies is a real challenge and grind of phases. It's 100% luck and persistence.
Source
I don't think Nintendo has ever commented on this game. I searched "official Pokemon rumble interview" and literally nothing came up.