Can each variant of a product have its own shipping cost?

Yes. Open the variable product, go to its variants list and click Details on the variant row. Along with barcode, weight and dimensions, there is a Shipping cost box for that variant. Fill it in for each variant and save the product.

These costs are used when Shipping Type is set to Per Product under Settings → Store Settings → Shipping Setup. In that mode each line's shipping is its own cost times the quantity.

In Overall Product mode, one flat cost applies to the whole order and variant costs are ignored.