Will be the discount applied if the bundle is added twice?

Yes, the discount can be applied to a bundle multiple times in a single order.

For example, you can see a discount applied 3 times in the following screenshot:

This is also documented at https://help.shopify.com/manual/promoting-marketing/discount-codes/discounts-faq#how-many-times-do-product-specific-discount-codes-apply-to-a-checkout-cart

Please note that the Shopify Discount system does not allow to apply two discount codes at the same time so, if you add two different bundles, only the second one will be discounted.