Does BYOB support stacking discounts on Shopify?

Yes, by default BYOB generates a new variant with the suitable bundle price and weight. Therefore, you can further stack a discount code or auto discount on top. For example, you set up a bundle to select 3 products for 10% off. The total price is $100. The discounted price is $90. There will be a new variant priced at $90. Customers can redeem the discount as usual at checkout

The discounts applied:

1. The bundle discount 
2. The native Shopify discount at checkout

Doesn't Shopify support combining discounts?

Combining discounts is different from stacking discounts.

The combining feature states: "If an item is eligible for multiple discounts, only the largest discounts with apply". If you set up 2 discounts that can be combined, only the best discount is applied, i.e. only 1 discount is used.

The discount applied:

1. 1 discount - the best one