I am very unfamiliar with the shelve aspect of Git (EDIT: not an aspect of Git, but rather IntelliJ IDEA feature of shelve).

If stash is used to put aside unfinished work what is shelve then? What would you use it for?

For example on Update Project (from VCS menu)

one will get (in IntelliJ IDEA 2019.2)

git shelve doesn't exist in Git.

Only git stash:

• when you want to record the current state of the working directory and the index, but want to go back to a clean working directory.
• which saves your local modifications away and reverts the working directory to match the HEAD commit.

You had a 2008 old project git shelve to isolate modifications in a branch, but that wouldn't be very useful nowadays.

As documented in Intellij IDEA shelve dialog, the feature "shelving and unshelving" is not linked to a VCS (Version Control System tool) but to the IDE itself, to temporarily storing pending changes you have not committed yet in changelist.

Note that since Git 2.13 (Q2 2017), you now can stash individual files too.