I want to assign the OOTB "Share" functionality of SharePoint 2013 to a custom button. How to a

check the below post you can change the share button by changing the PromotedActions Delegate Control and add your own button with your custom functionality  

http://zimmergren.net/technical/sp-2013-some-new-delegatecontrol-additions-to-the-sharepoint-2013-master-pages

I am not looking to edit the PromotedActions delegate control. I am creating a custom delegate control (contains button). Now, I want to assign the "Share" functionality to that button in custom delegate control.

The clientTemplates.js has code that displays the Sharing Dialog. This dialog is implemented in the Sharing.js. Both of these js files are in the layouts directory. If you have the Sharing.js file loaded on your page, then your code could possibly call this code with using the root url, list name, and list id.

function DisplaySharingDialogForListItem(renderCtx, listItemID) {
    EnsureScriptFunc("sharing.js", "DisplaySharingDialog", function() {
        if (typeof listItemID === "undefined") {
            var listItem = renderCtx.CurrentItem;

            listItemID = listItem.ID;
        }
        DisplaySharingDialog(renderCtx.HttpRoot, renderCtx.listName, listItemID);
    });
}