The point, to some extent, is that when you talk about a "hole" you need to specify what dimension you're referring to. Making that precise is hard, and is one of the things that makes serious topology difficult.

And useful. I've worked with robot arm systems with 9 degrees of freedom, and thinking of the configuration space as a 9 dimensional manifold with "holes" was the key point.

The terminology isn't misleading, it's technical.