After creating a support ticket with Shopify, this is what I found out. I copied and pasted the last email with them…
Thank you very much for all the screenshots you sent, I did some more digging in the matter for you.
What I can see is that the over 1000 products that were not published to Google all have the following error: You need to add a gtin or mpn.
Since your products are custom products, this warning is expected but reads incorrectly. This should be showing as a “limited performance” warning and the correct “Limited performance” warning can be verified in your Merchant Center account of Google. The product is approved and available for Google Ads. At this point you will not need to take any further action.
Please check the status in your Google Merchant Center account and check if this is all set away correctly. If this is correct then as explained above everything is as it should be and you do not need to take any further action. What I will do on my end is make a note for our developer and tell them about the message that pops up in your admin so that they can work on this issue being removed in the near future. The good news is, everything is published as it should be.
Thanks again for all the screenshots, that is very helpful. We have found out that the problem will not affect your Google Ads. As mentioned before I will make a note of this, to get the warning fixed.
—I did go into my Google Merchant Account Center and my products were showing as published, so this is a Shopify issue/mistake. I suggest you do as she suggested if you have this same issue and check your Google Merchant Account Center.