IDs in an HTML document must be unique. From your description, you're using the same block of HTML for multiple items, which is generating multiple elements with the same ID.
You'll need to change the way you generate and process your HTML.
<a class="like" href='@Url.Action("AddLike", "Like", new { item.Id })'>
<svg ...>
...
</svg>
<span class="like-count">@item.Likes.Count()</span> likes
</a>
$(document).on("click", "a.like", function(e){
e.preventDefault();
var me = this;
$.post(this.href).then(function(result){
$(me).find("span.like-count").text(result.likes);
});
});