[Vuejs]-Why does the PUT request give an error using fetch in Vue?

Your backend accepts CORS POST requests, but not PUT. You should modify the headers set there to include PUT using Access-Control-Allow-Methods response header.

PHP example:

header("Access-Control-Allow-Methods: GET, POST, PUT, OPTIONS");

More info here: https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Access-Control-Allow-Methods