How to fix syncing issues between Google Merchant and Shopify app?

For correct error and warning reporting only use Google Merchant Center > Products > Diagnostics.

When submitting product identifiers the following rule applies.
If a manufacturer has assigned a bar code, then submit brand, mpn (if available), GTIN and set identifier exists to yes.
If the manufacturer has not assigned a bar code, then remove the values brand, mpn, gtin and set identifier exists to no.
References:

https://support.google.com/merchants/answer/6324351
https://support.google.com/merchants/answer/6324461
https://support.google.com/merchants/answer/6324482
https://support.google.com/merchants/answer/6324478

Apparel Products

For apparel products go to Shopify > Products > Click on your product > More actions > Google Fields > add the variants