Hard to tell without pictures, but adding the Hybrid Wax did no extra damage, however, since you didn't wipe away the residue after spraying, you essentially just liquified the bird droppings and probably made it affect a bigger area than where they originally fell.
Bird droppings are very acidic and since it wasn't completed wiped off, the bird droppings most likely left some etching marks.
It is hard to tell if they will go away after a couple of washes so I can't say for certain, but since you already washed your vehicle, no extra damage will occur.
If you have dark-colored paint and an eagle eye like myself, you will most likely still see them after a few washes, but I would probably wait it out to see.
Hybrid Wax will have no effect on the marks because there are no abrasives in it to polish out any imperfections in your paint.
If you find they don't go away after a few washes, the next step would be to use a clay bar. Clay bar will 100% remove any surface contaminants so anything the clay bar doesn't remove is the damage caused by the bird droppings.
The only way to remove the damage after using clay bar would be to polish/compound the area. If it's a small area, this can be done with some polishing compound and a handheld foam pad.