Can I label my canned responses?

You cannot label responses. But you can link them with your organization teams. For example, here I created a team name Backend, and then linked a response to it.

When searching a response you can prepend your search with the name of the team (see image below). There I search “backend” and it selected the only response linked to this team. [team name + search query] will find only responses from a team containing the search query.