I just saw this thread which might be answering my question, but as I am unsure, I would like to clarify why という is being used here? Does anyone have any ideas?
という has various usages, and という in your sentence is explained in this answer. That is, 5個入り ("containing 5 pieces") is working as a explanation of ボリューム感. So a literal translation is "voluminous feeling of having-five(-in-a-package)!".
Although 5個入り happens to contain a number, your sentence has almost nothing to do with the
何 + number + という explained in the question you linked. This construction must be used always with 何 (e.g., 何十という人, 何万円という金額, 何千万キロという距離).
As a meaning of という itself, rather than a pattern, the one in question and the linked question (何百というふくろう) are the same:
It means something like as many as: as many as five-piece set (and as many as hundreds of owls)