Has anyone tried using a miles card to book accommodation or flights via Shopback? Will I still be earning both miles and "cashback" in a sense?

Short answer, yes. The miles goes to your account with the bank while the cashback goes to your shopback account.

But note that banks sometimes have promos with airlines and hotel websites and you must use the specific link to enjoy the additional benefits. But it means you will lose out on cashback, since you can't enter the site through shopback.

In this case, you have to weigh out the pros and cons before buying.

As long as you have read through the terms and Condition to get the cashback from ShopBack, you should be able to get both miles and ShopBack cashback from the booking (login in through ShopBack to register the purchase, and using the credit card without any other promotions from the merchant itself).
Please read properly the T&C.