I think you are both right to some extent.
God made a covenant with Israel, so every natural descendant of Israel is bound by it. In that sense, they are God's people, who are subject to the Law, its blessings and curses, but they will be redeemed from the Law and the curse if they believe in Jesus and die to the Law (Rom. 7:4). Not all of them believe in Jesus today, but one day all Israel will be saved (Rom. 11:26). They are still beloved for the sake of the fathers (Rom. 11:28).
But I have a suspicion that not all natural descendants of Israel are God's chosen people. This is dangerous territory, so please correct me if I am wrong. But this is my reasoning:
In Exodus 19:5-6 (NKJV), God says that the people of Israel would be a special treasure, a kingdom of priests, and a holy nation, but it is conditional on "if you will indeed obey My voice and keep My covenant". Fast forward to Deuteronomy 7:6 "For you are a holy people to the LORD your God; the LORD your God has chosen you to be a special people for Himself, a special treasure above all the peoples on the face of the earth." It is now an unconditional statement. But notice that Moses wrote Deuteronomy 7:6 after 40 years in the wilderness. All of the disobedient Israelites who refused to enter Canaan in Numbers 14, the rebels that followed Korah, Dathan and Abiram in Numbers 16, and the people who worshipped Baal of Peor in Numbers 25 had already died. Only the new generation of faithful Israelites were alive. Deuteronomy 4:4 says: "But you who held fast to the LORD your God are alive today, every one of you."
Centuries later in Hosea 1:9, God says to the people of Israel, particularly the northern kingdom, that "You are not My people, And I will not be your God." Of course God longs for the people of Israel to be His people, but it seems like it is possible for this relationship to be broken for individual Israelites, though not for the whole nation. As Aaron56 quoted from Romans 9:6: "They are not all Israel who are of Israel". Also Romans 11:5 says "Even so then, at this present time there is a remnant according to the election of grace". Elect = chosen. So those who are elect/chosen and believe in Jesus are God's elect/chosen people.
I think 1 Peter was written to Jews in the Dispersion who believed in Jesus (1 Peter 1:1-2). 1 Peter 2:9-10 says: "But you are a chosen generation, a royal priesthood, a holy nation, His own special people, that you may proclaim the praises fo Him who called you out of darkness into His marvelous light; who once were not a people but are now the people of God, who had not obtained mercy but now have obtained mercy"