I let my kids choose their own instruments for school band. For private lessons that I pay for, I pick the instruments, with input from the children.

No to the same instrument. There will be discord! Kids learn at different paces, they will compare themselves to each other, it's just not a good idea. If music is the priority in your home, then you need to let your kids choose their instruments and find a good teacher for each of them.

If money and/or time is very tight, yes. Otherwise, let the kids choose.

If you have or had multiple children in elementary/middle school band and supplemented with private lessons, did each child select his or her own instrument? Did you encourage all of them study the same one, in order to streamline the private lessons (schedule with one teacher, etc)?