Consuming protein through out the day on training and non training days will allow your body to state in a positive nitrogen balance. This is important for tissue repair and growth.
So yes, you can have a protein shake on non training days. Protein shakes are not necessary... even on training days... but shakes make it convenient to consume adequate protein to support your goals.
Protein shakes are also portable making it easier to consume enough protein even with the busiest of lifestyles.
Remember... protein shakes are supplements and they are not meant to replace food.
Rule of Thumb we Give to Clients
If you eat 3 times per day ... 1 of the meals as shake is OK.
If you eat 4 times per day... 1-2 of the meals as shakes are OK.
If you eat 6 times per day... 2-3 of the meals as shakes are OK.