Hi Florian,
Thanks for reaching out to us!
Regarding how to customize this widget, take e a look at the Customize part of the how to configure article. This Week widget doesn’t come with the option to show the same tooltip as the week view has, while this is possible via a customization it’s not supported at the time. I guess you can re use the tooltip template from week view in some way, but you’ll need to try that out as we don’t have any guide or process to do so.
You are welcome to suggest this as a feature in The Events Calendar user voice page.
Please let me know if there’s anything else I can do for you,
Best,
Nico