Reinbacher might be saying he models his game after Josi's but if you look at both of them on the ice, their playstyle is actually quite different so I don't see the direct parallels there aside from nationality.
Put simply, Josi is a modern offensive defenseman that has ++ skating, poise, good puck-possession skills, is amazing in transition, as well as owns great offensive skill carrying the puck and holding the blueline on the powerplay. And although Josi is far from a slouch defensively, he is also not anywhere near a shutdown defenseman.
Reinbacher on the other hand, is much more defense-first than Josi, slower, less skilled, less adept of a shooter, and much more raw in his puck carries, meaning that his offensive potential is much lesser than Josi's. Reinbacher's decisions with the puck, although fast and reasonably mistake-free, are also not nearly as creative offensively.
But Reinbacher is also a LOT better defensively, has better core stength, much tighter gap control, understands angles, positioning and passing lanes a lot better in his own zone, fares much better against the cycle, is a better battler for the puck along the boards or in front of the net, and just generally a lot more polished when it comes to defending than Josi was at the same age.
So again, I personally think Reinbacher's game is much more reminiscent of Jaccob Slavin than it is Roman Josi, no matter what he says/yearns for in interviews.
But hey, I'd love for Reinbacher to prove me wrong and actually go on a Roman Josi-esque development in the NHL. That would be amazing.
I just don't see it though.