The problem is, the last time a common genetic ancestor we share with mice was alive, dinosaurs roamed the Earth. Like a lot of promising starts (Interferon comes to mind) it remains to be seen where this goes...
Thus a molecule that binds to the XYZ receptor in mice is very likely to bind to the XYZ receptor in rats and dogs and humans - be metabolized by the same enzymes into the same products, etc.
And all of this is checked before the therapeutic goes into humans. Does it bind to the human version of the target? Do the metabolites produced in model species correspond to those that would be produced in humans (using human liver cells), etc.
People wonder why monkeys are used in drug trials - because they are much more likely to be a relevant model.
But it is amazing that there is so much similarity in the active working parts of proteins between mice and humans that they are useful at all!