Is git stash branch-specific or for the whole repository?

No and No. git stash is per-repository.

Here is a nice page on how to use it.

To see the current stash stack:

git stash list

To pick a specific stash from the stack, refer to it by the stash@{number} shown by the above.

If you want the behavior to be per-branch, you can just make a commit (or multiple commits) on the branch. You can always "unmake" the commit(s) later (e.g., with git reset, either --soft or --mixed; see the git reset documentation; or with git rebase -i to keep only the eventual "real" commit(s) while discarding the temporaries).

(To really emulate git stash you need at least two commits, one for the index state and one for the work-tree state. If you're not planning to save and restore the index state, though, you can just git add -A the entire work-tree state and put that in the temporary commit. Alternatively, git stash is a shell script so you could copy and modify it pretty easily to make it work per-branch by default, using, e.g., refs/pb-stash/branch as its working name-space, rather than the single global refs/stash for the entire repo. You'd still be able to bring a stash from one branch to another by naming it explicitly.)