OK. Did a quick bit of research and while the two bridges look nearly identical, they are slightly different.
Jaguar Bridge:
Mustang:
And yes, a lot of guys seem to swap the Jag (and the Jazzmaster) bridges for the Mustang ones, and it apparently cures the problems you mentioned. I also know a guy (himself a luthier) who likes using the Graph Tech Tusq Mustang saddles on all of them: Jags, Jazzmasters and Stangs (he has quite a few of all three and also the offset Fender VI bass).
But, I've never done it myself....