The dependence of learning generalization bounds on the dimension of the instance space
I think the VC-dimension term pretty much takes care of it. You can think of three cases:


The hypothesis space $H$ is far more "complex" than the input space $X$
$H$ and $X$ are matched up well to each other
$H$ is much "simpler" than $X$


We are usually only motivated by the second and third cases, so the VC-dimension term in your bound captures what we care about.

For example in $X=\mathbb{R}^d$, you probably want to choose a hypothesis class that depends on $d$, for instance halfspaces in $\mathbb{R}^d$, and so $d$ shows up in the VC-dimension term (in the case of halfspaces, this is $d+1$).

There are also some examples of classes with finite VC-dimension that are still interesting. Here even if $X$ is "large" or complex, we can get a small bound.