Not quite everything. Since Mercurial branch names are baked into the commit, if you wanted to know which branch a commit was originally part of, you can find out.
Git does not support this. If two branches have parent commits in common, it technically belongs to both branches.
There is a valid conversation to be had on if this feature is actually necessary or makes sense in the context of a DVCS, but git branches are not a full substitute for Mercurial branches.