Hi, Cheese Cake!
I tighter embouchure is pretty much never a good idea. It's very easy to use your embouchure (with strong, fast air support in place) to bring the pitch down, and in doing so it can actually open up the player's sound to make it more flexible and full. As saxophonists get older and more experienced, they very often find themselves pushing their mouthpieces in more and dropping their jaws more since that offers them the flexibility to play in tune more often. The saxophone is naturally a VERY out of tune instrument so a lot of responsibility lies on the player to listen and adjust, and having a flexible embouchure is how to make that happen. If this mouthpiece is all the way in already, the saxophone might need some adjustment (key height can affect intonation). Two other possible things to check out would be reed strength (moving up a half strength might bring pitch up a bit) and air support.
If a saxophonist's air support and tongue position are in good shape -- that is to say, air speed is strong from a lot of support from the stomach muscles, even at low volume levels, and the tongue position is high in the back with a strong "EEE" syllable -- he or she should be able to bend down an entire step from high "B" down to the "A" below it. If that isn't possible, then some work needs to be done on air support and tongue position, and doing that will help intonation and sound production a lot.
If she plays on another student's saxophone and pitch is much better, then sadly it might indeed be the horn. Student Selmers aren't very consistent and don't have a great reputation for good workmanship. I student or intermediate Yamaha will probably be much more reliable!