New versions of programs can sometimes be found in the official packages of newer Ubuntu releases.
For example, the DAE-software Asunder gets installed in version 2.8-3 in 16.04, while the most current, fully-updated version in 15.04 is still 2.7-2.
Is this because 15.10 will lose support in a few weeks or do LTS releases also not always receive the newest versions of featured programs? If not, is it possible to update the newer packages somehow?
No package will typically be updated beyond the major version shipped with the initial release of the distribution. Minor updates may be released provided changes are not significant, and security fixes from newer versions may be back-ported to maintain compatibility.
You will find that some packages independently offer newer major versions from their own repos, but that's at their discretion and those packages wouldn't be supported by Canonical.
