Sorry for taking a very long time to reply.
Google Maps API key is required to make any calls to Google Maps API. Without an API key included in the API calls Google servers reject all API calls and maps cannot be displayed. The API keys are free, but you have to create an account and register a payment method since use of some API calls is billable (if the usage exceeds a call-specific limit. Some basic API calls are free. Custom Maps uses only free API calls. In addition to billing (and maybe more importantly) Google uses the API keys to limit API abuse. If they detect some API key being used in a way that doesn't meet their terms and conditions, they can disable the API key and possibly contact the owner of the key.
Google Maps API is used by Custom Maps only to display the basemap (and the overlaid map image and tiepoints) when a map is being created. Google Maps API requires network connection, so it is not used when the created custom map is being displayed.
Google Maps API is also used in an anonymous way. Due to use of the API key, Google knows who is responsible for the program being used (the creator or current owner of the app, in this case they are both me). But no personal information about the Custom Maps user is sent to Google servers. And since Custom Maps doesn't even ask for any such information, it wouldn't be able to send it anywhere anyway.