No, I can say that these two do not always go hand in hand. You have stories that is dark but have no depth and stories which are light and have plenty of depth.
What makes a deep story is how the author uses the themes, the setting the characters to give out depth. To make layers between simple tropes, plot points and small details come together to weave an intricate story that can’t be unravlled unless you have read it all. It is making the reader think without been even realising it.
A story can have dark themes but it isn’t always good. It could be there to merely add something without really understanding or exploring what it means. Or really understanding the weight and delicate nature of the topic.
Things like prostitution, gambling, addiction, drugs, violence. Are all things we mark as dark. But a lot of them are not employ in such a manner that makes it deep or thought provoking. It is merely an aspect used to make it seem more adult like without truly understanding what it means.
As such, a deep story is one that uses all the details of it, to make a compelling and sometimes penetrating message without the viewer knowing. Such as how my own works, with A Maiden’s Strange Voyage touching a lot on the idea of freedom and choice, in a time where people did their duty without any concern.
As such I feel that stories which are dark are not necessarily deep, and light stories can have very very dark meanings. (An example will be spirited away, which happens to be a story about prostitution. Given how Chihiro is given a new name, it does seem likely.)