Fortunately there is a utility called lsb_release that provides this information:
bash$ lsb_release -si
bash$ lsb_release -sr
bash$ lsb_release -sc
Complete all the way down to the silly nickname for each release.
Since this is an LSB utility, it's probably been in place for some time, silently waiting to provide more detail than uname. Seriously, would a mention in the SEE ALSO section of the uname manpage be *so* hard to add?