It's much less expensive, and more common to bolt them to blocking (a stud can work, but you won't be able to get all of the screws into one and they are rarely where you want them!). The nice thing about the WingIts, is that they provide more than the ADA requirement for a safety bar and are designed NOT to go into studs or blocking. In fact, they can't be too close to blocking or a stud, or the wing on the back won't expand and it won't work (if it will go in at all). There's no reason why you couldn't have one end in a stud or blocking, and the other use a WingIts fastener, though, if that worked out better. They give you the flexibility to put the bar pretty near anywhere, and still provide the strength you need.
While they do sell safety bars as well, IF the screw locations on a different bar are the same (the all SS one has several bolt patterns, the one in the left picture requires holes at the locations of the screws shown), they will work just fine. The bar I installed with the SS fasteners was not a WingIts bar, those I installed with the plastic version were. I was really impressed with their tile drill bit...it didn't walk at all, and cut through the tile quickly and cleanly. The tile was fairly soft. On the first one, it was a hard porcelain, and I used a diamond hole saw (I don't think they had their custom drill bits available when I did that one). I feel fairly confident it would have cut through that as well, but probably more slowly and it would have worn it out faster (they claim a life of about 200 holes on their bit).