Generally, yes you need GTINs.  Usually your Barcode field in Shopify is mapped to GTIN in Amazon.  Amazon uses your GTIN to identify the product you are selling.

Exceptions for requiring GTIN

1) If you are only creating an Amazon Offer (product already available in the Amazon catalogue 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)