Before using the New Product function and page, you must first visit the configuration page for the Site that you want the product to pertain to.
Stores > All Stores > [select Web Site] >
On that config page, check the Set as Default button.
Now start the Add New Product process.
Way down the Product configuration page there's a setting for "Product in Websites", on which you can set which Websites the product is available on. (And not to be confused with the Categories setting and mechanism.)
It looks like the "Set as Default" (Website) preselects that Website in the "Product in Websites" settings.
You can use "Product in Websites" to add one or more additional sites. After saving that setting, the product config page will then display a Scope control at top left, including scopes for those sites.