The page in the iframe could make an AJAX API request to "https://[
your.canvas.com]/api/v1/users/self" to determine the logged-in user. You will get their name, SIS ID, SIS Login, Canvas ID and some other info.
function reqListener () {
console.log(this.responseText);
}
var req = new XMLHttpRequest();
req.addEventListener("load", reqListener);
req.send();