However the company does not claim copyright or patents on work-unrelated ideas and (closed-source) projects.
Do you get fired, or do they claim ownership?
I doubt that would happen in practice though, unless you were creating an open source competitor to something they do or somesuch.
Having to worry about it is horrible enough, since this will actually keep many developers from pursuing their passions in their free time.
I would never agree to this. Generally, if you work at a place that commonly throws around the phrase "<X> is in breach of contract which is grounds for termination.", you're in a bad job that will not ever teach you anything or advance your career. You are working as a glorified contractor and should learn some warning flags to avoid the terrible corporate bus stops.