I guess there are multiple names, but I know it as the river's "long profile", and it is used in academic papers (one example here and a search in google scholar here). But you can find similar diagrams under names like the river's height profile, longitudinal profile, longitudinal map or geographic profile. These last two are the ones used in an Australian government web page that have many of those profiles for all major rivers in Australia, with the cities along the river and other information. I'm not aware of a global repository of long profiles.
The following picture is from the same link pointed above (from Geography AS Notes)
A quick search for "Danube long profile" put the following image in the second position of the results
Taken from here