Do I need a GTIN for custom-made items on my Shopify store?

You don't need to create GTINs for custom made products. There is no need to invest money into this process. SKUs (Stock Keeping Units) are used for inventory management and it is always recommended to generate and assign SKUs to your products. Make sure that you generate your SKUs in a systematic way. When you use Google Shopping it is important to have your product information properly set up. It will improve the performance of your campaigns. Here is a good post with an overview about GTINs - https://keepshoppers.com/forum/technical-issues/unique-product-identifiers-requirements-for-google-shopping

It's not mandatory to have Shopify GTINs for your custom-made items. You can use SKUs instead of Shopify google GTIN. However, it is recommended to generate GTINs as it enhances your ads and listings performance and makes the inventory tracking more manageable than ever.