Can doctors' offices save part of the syringe for touch ups?

The reason some surgeons will not re-use leftover filler in a syringe is because as soon as you open the package, the contents are no longer sterile. Ideally you use the filler in one sitting. That being said, many surgeons will save leftover portions for a touch-up, and those that do will usually remove the used needle from the syringe (leaving a used needle attached would risk bacteria from your skin going through the needle and back into the syringe) and replace the sterile cap, and seal the package. However, this does not mean the package is once again sterile, as this was lost as soon as it was opened. If the decision is made to use the leftover for a touch-up, this means both the surgeon and the patient accept the risk (however small the risk may be, but is still present), that during the time that has passed since opening the syringe, there is possible bacterial translocation involved, and a potential risk of infection. The longer the syringe sits after the package was opened, the greater the theoretical risk.

It depends on the practice.
We sometimes save part of the unused filler in case the client wants more later (ie. for lip augmentation) but this will vary from client to client.   When we do it, we let clients know that it is against manufacturer recommendations and that anything longer than 1-2 weeks is not ideal.
Ultimately it's best for you that the filler is fresh, pliable and not contaminated.