Interpretation of a PhD degree in a very broad, interdisciplinary field
I completely understand your feeling. I can tell you that many other people in this field feel the same. This has nothing to do with how smart or how good they are. This issue also bugged me a lot, but I can offer some insights:


Bioinformatics/computational biology is really huge and you cannot be an expert in all aspects. Even if you look at senior scientists in this field, I do not think there is someone who is an expert in all subfields. You simply can't master all the physics, math, CS, chemistry and biology at an expert level - even in a much longer time than a PhD.
You can still have a huge impact without being a super-expert in every subfield. Just look at some of the research published in top journals. The reason is that you will have knowledge and a way of thinking that people restricted to a single field may not have. From personal experience, I can say that this is a significant advantage for asking certain types of questions and coming up with certain ideas that single-field specialists won't come up with.
After a while you will realize that you actually are an expert. Maybe not in the sense that you know everything about everything, but you will see that you can give good advice to other people, foresee potential problems, and so on. In addition to knowing a lot about computational biology, you will become an expert in things such as quantitative modelling, applied machine learning and "big data" analysis (I hate that term), skills which are very useful in a wide range of fields.
The fact that you cannot become an expert in everything doesn't mean you should neglect learning. On the contrary, you should constantly try to expand your knowledge in all related fields. And yes, it can be more difficult than learning only one subject.
Finally, in the end you will be working on a specific problem in a given biological problem with a given set of tools. That problem is what you really need to be an expert on.