The path could be pretty long. The dopesheet and graph could be smart and only show enough parent folders to make the names unique. That way most rows won't show folders, unless needed to disambiguate. We do that on the file open dialog for the project names shown there.
I worry that one or two parent folders may not give enough context in some cases, even though it results in different names. Maybe when there are multiple names, we could use the full path.
Note this also comes up for deform timelines, when different slots under the same bone have attachments with the same name, you can't tell the timelines apart.
