Will be the discount applied twice if the bundle is added twice?
No, only a single discount can be applied to a single order.
If a discounted bundle is added to the cart twice, a discount will only be applied to one of the bundles. In the same way, if two different discounted bundles are added to the cart, only the discount of the last one will be applied.
For example, in the following checkout image, where a 10% discount has been configured, you can see a $100 bundle added three times and how the discount ($10) is applied to just one of the bundles.
Here we are another example where the "Cheapest item for free" discount has been configured and a bundle has been added three times, you can see how just one of the cheapest products is free: