yandex-maps service for save places
You should be using function such as:

localStorage.setItem('points', JSON.stringify(points));
localStorage.getItem('points');


If you can use es6:

// instead of 
var coords = e.get('target').geometry.getCoordinates();

var pointLat = coords[0],
    pointLng = coords[1];

// Could write  
const [pointLat, pointLng] = e.get('target').geometry.getCoordinates();

// Why save points as variable if it's only used once
var points = getPoints(),
    newPoints = [];

newPoints = points.filter((point) =&gt; point.lat != pointLat &amp;&amp; point.lng != pointLng);    
// Could just do
var newPoints = getPoints().filter(point =&gt; point.lat != pointLat &amp;&amp; point.lng != pointLng);