Logically, fooA contains a bar, on the other hand fooB does not contain one because the bar is not registered. Does the child containers behave like this ?

What he said.

I second Moutasem's opinion.

Both should return a Foo without a Bar.  If a Foo with a Bar is returned and childA is disposed (and with it the singleton Bar), then Foo will be left referring to a disposed object.  Another problem with returning Foo with a Bar is that if childB is resolved first then the behavior will be different and having different behaviors based on the timing of calls to Resolve is not acceptable

One possible solution would be to add a lifetime of "singleton-per-container" in which case childA would hold its own Foo with a Bar and childB  would hold a different Foo without a Bar.

I've played with child containers and Unity and run into a number of subtle issues mostly having to do with singletons.

I agree with your assessment in the original post: "removing the child containers, and add some nicer support for contextual scoping of components would be the best solution"