Do Gentiles have the same 'My people' relationship with God that Israel had?
No. Scripture indicates Gentiles do not enter the same intimate "My people" relationship God had with Israel. God does call out from the Gentiles a people for His name, but Gentile believers are more correctly referred to as His born ones. Israel will again be called God's people when restored in belief at Christ's return.
In Acts 15, after Peter quieted the opposing Jews, James answered and said Simeon had declared how God at the first visited the Gentiles to take out of them a people for His name. Then verse 15 agrees with the prophets, quoting Amos: after this calling out of a people for His name, God will build again the tabernacle of David, fallen down, and set it up. The conclusion was made that yes, God is going to call out of the Gentiles a people for His name. However, Les teaches that scripturally Gentiles certainly do not come into that intimate relationship God had with Israel when He called them "My people." When Israel fell away in obstinacy and unbelief, God said to Moses, "Thy people." It will not be until Israel comes back in complete belief, when the return of Christ has taken place, that they will once again be called God's people, or "My people." But other than this portion in Acts, where God says He will call out a people for His name, Gentiles are more correctly referred to as His born ones.
Simeon hath declared how God at the first did visit the Gentiles, to take out of them a people for his name.