I saw on a few websites that they have different products on the front end for different colors, but once I click on it I see that it’s the same product just showing each color as a separate product on the category page.
The different colors for the same product are added as variants. You can add options on the product listing page in Shopify and add colors and enter the details.
You might also check the AdNabu for Google Shopping, which can help you send the correct information to your merchant center feed and update it automatically.