You mix objects and strings in your code. Your is album_contentinitialized in the loop as an object ( {name: item.name, uid: 1}). On the next line, you treat it as a string, adding a comma.
In general, use an array to collect all of your objects. It is (almost) always better to use your own objects, rather than trying to mimic them somehow.
.
var albums = '';
var album_list = [];
$.each(data, function(i,item){
var name = item.name;
albums += '<li>'+item.name+'</li>';
var album_content = {name: item.name, uid: 1};
album_list.push( album_content );
});
localStorage.setItem("albums", JSON.stringify( album_list ));
var albums_l = JSON.parse(localStorage.getItem("albums"));
$.each(albums_l, function(i,item){
console.log(item.name);
});