Generally, yes you need GTINs.  Usually your Barcode field in Shopify is mapped to GTIN in Amazon.

Amazon use your GTIN to know what product you are selling.

Exceptions from requiring GTINs

1) If you are only creating an Amazon Offer (you are selling something already available in the Amazon catalog sold by other sellers) then it is possible to provide Amazon with the Amazon Standard Identification Number (ASIN) to tell Amazon what product you are selling.  You can paste this into the ASIN field in Amazon Channel by Codisto (pasting an Amazon product URL is usually sufficient to get the ASIN).

2) You have a GTIN exemption (learn more)