The Old Testament is filled with type and shadow, meaning that the full revelation of truth is yet to be revealed through Jesus Christ. This is why Jesus Christ often corrected misunderstanding from the Old Testament. The people of Israel were God’s covenant people in the earth. God gave them many promises, including that of making them a kingdom of priests.   

and you shall be to Me a kingdom of priests and a holy nation. These are the words that you shall speak to the sons of Israel. -Exodus 19:6

In the new testament we see these words written again by Peter, but now we have the full revelation, which is why Peter says more than what Exodus 19:6 says.
 
But you are a chosen people, a royal priesthood, a holy nation, a people for God’s own possession, so that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light; - 1Peter 2:9

The people in covenant with Jesus Christ, both Jew and Gentile are that kingdom of priests and not the nation of Israel as a whole. The kingdom of priests is made up of a remnant from both the Jew and the Gentile.