Your vet may have meant you to draw up the total dose you need overall, and change the needle for each animals. We do this with the sheep on worming day- draw up with one needle, change it for the sheep, use the same drawing up needle, new needle for a sheep- etc, etc. This is common in large animal practices. I wouldn't reuse the needle the next time, though or anything. If you are using a sterile syringe, and a sterile needle for each fill, there should be no probelm- we have never had any.
Before disposable syringes and needles were so common and cheap, glass syringes and reusable metal needles were common. The syringes could be either autocalved or heated in an oven at 80 degrees C for 48 hrs, the needles were heated in the oven.