Is their any specific method to store miswak after its use.
or it can be stored same as brush.
Ref. No. 1000/41-190
In the name of Allah the most Gracious the most merciful
Then answer to your question is as follows:
The way to keep a Miswak is to keep it upright, not horizontal on the ground. ولایضعہ بل ینصبہ والا فخطر الجنون۔ (درمختار مع الرد1/235
And Allah knows best
Darul Uloom Waqf Deoband