Hi Leila,
It's not a Stripe setting, and it's a bug on our side. Thanks for reporting it.
HikaShop gives Stripe the address to come back to, and that address does carry the menu item your checkout runs on. But when the customer came back, the plugin sent them on to the end of the checkout without that menu item, so Joomla displayed the page on the default menu item of the site, your home page, and you got its content around the thank you message and the order link.
This is fixed in version 1.5.4 of the Stripe Checkout plugin, which you can download again from your account on our website. After that, the customer comes back on the menu item the checkout was on, with its own template style and its own modules.
If you want that page to look different from the rest of the shop, create a menu item of type HikaShop > Checkout, in a menu you don't display if you prefer, and select it in the "Force a menu on the checkout" option of the Checkout tab of the HikaShop configuration. The checkout and the page after the payment will both use it, and you can assign your thank you content to it.