They anointed David king over the house of Judah: this they did upon just grounds, because not only the kingdom was promised to that tribe, Genesis 49:10, but David was designed and anointed by God, whose will both they and all Israel were obliged to observe and obey. And they piously resolved not to neglect their duty, though they saw the other tribes would. Yet their prudent caution and modesty is observable, that they make him king of Judah only, and not of all Israel. And therefore there was need of a third anointing to the kingdom over all Israel, which he had 2 Samuel 5:3. But as for that first anointing, 1 Samuel 16:13, it was only a designation of the person who should be king, but not an actual inauguration of him to the kingdom.
Related Commentaries of 2 Samuel 2:4
2 Samuel 2:4
4 And the men of Judah came, and there they anointed David king over the house of Judah. And they told David, saying, That the men of Jabeshgilead were they that buried Saul.