I have used a regular vise many times on a couple of drill presses and on a milling machine. Because sometimes you just need a bigger vise.
I wouldn't recommend it for precision work, mostly because bench vises are designed to be able to clamp a wide variety of objects rather than to easily reproduce particular clamping configurations. So expect to spend significantly more time re-registering your work if you have to move it. Of course, that whole "clamp a wide variety of objects" thing has its advantages too, and has been the reason behind most of the times I found myself using one.
Also, be aware that most bench vises aren't particularly square. Don't assume that the top of the jaws are parallel to the worktable. Most of them are canted slightly to one side or the other. On some of them it will change based on how tight the rotation bolt is. (Just generally, they don't tend to be very precise implements.)
All that said, sometimes big, deep jaws just make things enough easier that it's worth the extra fiddle-factor to get the alignment right. For a drill press like the one pictured, I recommend using carriage bolts where the square section matches the width of the slots in the table. From there either drill the holes in the vise's flange bigger or turn down the shafts of the bolts and re-thread them as necessary depending on what you have.