Have you overridden the tab using a Lightning component bundle or some other way? Is the view overridden through a Lightning record page or some other way? Are they both supposed to redirect to Skuid pages? Are you going to the custom object’s tab through the classic app (it’s included in it), but it isn’t going to the Skuid page you expect? How do you want the app to work?
As for adding a non Skuid component to the page, you’ll have to use the Lightning app builder to start with a Lightning page and then add Skuid (and non-Skuid) where you want it. Here’s a general overview of using Skuid in Lightning.