You can copy all of the files, but most are not needed for the average user. bitcoind and bitcoin-qt are the main binaries. They are each self contained and do not rely on each other or any other files in the release.

/bin has all of the executables. Besides bitcoind and bitcoin-qt, bitcoin-cli is the only binary that might have meaningful changes between releases that you'd want to also copy it over. But generally, it's fairly version agnostic. The other executables in /bin are just utilities and the unit tests which are really only interesting to a very small set of people who probably aren't asking this question.

/include and /lib contain files for libbitcoinconsensus. If you don't know what that is, you don't need those files.

/share has manpages and a script for generating rpc credentials, most people don't need these either, unless you like using man instead of -help.

The easiest way to install is to just copy the folders into /usr, and do the same for every upgrade. That will copy every binary over, and also overwrite any existing binaries with the new version when you upgrade.