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You need to parse the string returned by PHP to JSON.
Add the following line
$.post(user_obj.ajax_url, request).always(function(response){
response = JSON.parse(response); // Add this line
console.log("This is inside vue" +response[0]);
console.log("This is inside vue" +response[0].title);
app.websites = response;
});
Source:stackexchange.com