Not only is that possible, but we do it all the time.
- No one can see the apps you register at stackapps.com/apps/oauth/register except you (and the SE devs).
- If you don't need OAuth, then you can enter just about any fake information in the required fields.
This still allows you to use thekeyto get 10K quota.
It is only when you want to access private_info or write to the site that you need OAuth and a public post here.
See, also: My app has to be published first? But it's still under development
If you do need OAuth, your app/script post here does not have to link to your domain (if any). So the public need not know anything other than that you have an app.
But beware that there has been a backlash against stagnant or malformed app/script listings.