It is a misconception that exercise should stop just because one limb is injured. Even if one part of your body is out of action, doesn't mean your whole body should be too. There are plenty of exercises you can do even if one arm is in a sling or your ankle in a moon boot.
Exercising on your uninjured side can also help the affected side in the long run. Research has shown that there is a crossover effect when exercising your uninjured side which can help prevent a significant loss of strength to the affected limb. This means that you when you have recovered the loss of muscle strength won't be as dramatic.
Exercising the uninjured side can also help to understand the movement pattern and exercise that you will be doing once your injured side is ready. You may be able to do one or two specific movements in your sling, but once out of it, your exercises will be able to progress much further. To help with this process, starting those progressions earlier on the non-affected limb can help with both your understanding and your muscle's ability to adapt to the exercise which will help reduce the overall loss of strength in the injured side.
This is where talking to your physiotherapist or exercise physiologist is really helpful. Not only can we provide you with exercises most helpful for your injury, but we can also help keep you motivated. No one wants to be injured, but it's important to keep moving!