There is an option to show/hide the title of a custom field. It works well on the productdetails page, I can hide it if I dont't need it, but it shows always in the cart and the order confirmation emails.
Is it possible to use the field's settings here?
No. You have not such setting.
If you need more flexible cart customfield see https://shop.st42.fr/en/products/shortcodes.htm and in the doc https://shop.st42.fr/en/shorcodes-english-doc#cart_persons
You can set option to hide/show the title in cart and/or in the mini-cart and many more...