Reuse sub-components in UML component diagram

The diagram you are showing is the notation for packaged Elements. Packaging has no semantics. You can move your elements wherever you want in the model browser and in the diagram. It only helps you, to keep the model nicely ordered. For this purpose it makes sense, that each element can be contained in only one packaging element. Therefore, it was not possible to do show the door in both rooms.

From what I understood, you want to model, that both the Bathroom and the Living Room have a part, that is of type door. For this purpose UML uses a composite structure diagram. It allows to show the internal structure of both components and classes. In this example, the composite structure diagram is shown in a compartment of the House component, which is shown in component diagram. As you see, the sub components of the House can also show their internal parts. This works for any level of nesting. Of course, if the diagram becomes too big, you can also use dedicated composite structure diagrams, that only show the internal structure of one component.

Now, some people call everything with a colon an "instance". In a way, they are right. The semantics of every structure diagram is that it tells us what allowed instances are - and that actually does not depend on the colon. Only here the diagrams tells us something about instances being a part of some other instance: Two instances of Door being a part of one instance of Bathroom and Living Room, who in turn are part of one instance of House. The modeler chose not to show, of what type of instance the House is a part. It could be a City, and the City could be part of a State, and the State could be part of a Country and so on. It really depends on the focus of the diagram (and the purpose of the model). The shown diagram only says something about all instances of the House, no matter what it is a part of. Therefore, there is no colon on the top level.

I didn't model this with Enterprise Architect. However, since this notation is firmly within the bounds of the UML specification, it should be possible.

I assume you are talking about software components. If you are talking about physical rooms, you should consider using SysML, which offers more possibilities to model the internal structure of things.