It looks good for a start! Of course, lots of work to get all the info there and up to date.
The left panel contains some references to external links, such as the glossary or user guide. I’d think such should be somehow indicated that clicking those leaves the doxygen page.
What would make sense is to also have various examples in this doxygen documentation, because those examples would then directly link to the respective function documentation, and the functions itself can also list the examples where they are used.
Just as comment to the prominent paragraph on " C++ Developers using HDF5 C-API functions beware:" This text seems to refer to the usage of exceptions within C++ code and particularly in callback functions. That seems implicitly clear, but I guess that this should be stated more prominently since usage of exceptions is not that all common (but of course it is in the HDF5 C++ API, even overly common there). It should be rather clear to C++ developers that C function callbacks don’t support the C++ exception stack. Another issue may be concerning C++20 coroutines… I am not sure how that would perform if a callback function yields execution to another routine which also does HDF5 calls, as this is not exactly same as multithreading. Something for future investigations, just to mention it at this occasion.