MPN
An MPN is the part number a manufacturer assigns to a product. It identifies the item within that manufacturer’s range, and unlike a GTIN it is only unique when paired with the brand.
It is the fallback identifier where no GTIN exists, and it is weaker: two manufacturers can use the same part number for unrelated products, so brand and MPN together are the unit that identifies anything.
Publishing both where both exist costs nothing and widens the set of situations in which your listing can be matched to the same product elsewhere.
Why it matters to a store
Without any identifier you are not a competing offer on a known product, you are a separate unknown with a similar name. In categories selling identical goods, that is most of the competitive position.