Assuming that your script localization and other stuff are set up properly, there is an issue in your back-end script. Your data is in the following format:
data : {
action : 'renove_vocabulary',
id_vocabulary : vocabulary,
id_user : user
}
But you are trying to get the data as follows:
$id_user = $_POST['user_id'];
$id_vacabulary = $_POST['vocabulary_id'];
These 2 should be changed to match the above. So, change them into:
$id_user = $_POST['id_user'];
$id_vacabulary = $_POST['id_vocabulary'];
Also, don’t forget to use nonces, sanitize the inputs and do some security checks. I would recommend using the REST-API altogether.