My God and my Lord is more of a phrase combining two names for God than a name of God in its own right. The phrase contains a form of the word ’Elohim and ’Adonai.
Hebrew Words for my God and my Lord
Strong’s Concordance numbers: 430, 136
Bible reference: Ps. 35:23
Stir up thyself, and awake to my judgment, even unto my cause, my God and my Lord. (Psalm 35:23)
A similar name (with the Hebrew words reversed) is found in Psalm 38:15 (and about forty times elsewhere): Lord my God (’Adonai ’Elohai). LORD, my God (Yahweh ’Elohai) is a name in Psalm 13:3.
Names of God that include both English words “Lord” and “God” can be confusing. The best way to distinguish them is by looking at the different Hebrew words that underlie the names. Both ’Adonai and Yahweh are translated “lord” in English but when Yahweh underlies the word, it is written in capitals “LORD.” ’Adonai is written “Lord.”
In the New Testament, when Thomas met the risen Jesus, he exclaimed “My Lord and my God!” (John 20:28)