Yes, you are correct that the product information should sync between Shopify and Google Merchant Center, including the product titles. When you make changes to your product titles in Shopify, the updates should be reflected in Google Merchant Center as well.
The synchronization process between Shopify and Google Merchant Center may take some time. Generally, it can take up to 24 hours for the changes to be reflected in Google Merchant Center. However, this duration may vary depending on various factors, such as the number of products being synced or potential delays in Google’s system.
If you still face issues after waiting for 24 hours, you can reach out to Shopify support or Google Merchant Center support for further assistance. They can help you identify the cause of the problem and guide you through the troubleshooting process.