Want to delete an old Facebook account

You need to straighten out if it's an account or a page.

If you can get to it from your own Facebook account, without having to log out and log in, then it's a page or a group and not a separate account.

Usually there is a gear wheel icon to the right (note: Facebook changes up its look all the time, so the specifics here may not be perfect) - check it when you are on the actual page you want to get rid of. You'll - if you are an administrator - be able to see some sort of verbiage to delete the group or page.

If not, then you'll possibly need to speak with whoever is an admin for the page.

But I'm not even so sure that's your issue. Could it be that this unwanted page simply 'liked' your own? If so, you don't have to show that on your page, but you probably can't just get rid of it.