The word "Lord" is used in a number of different ways in Scripture, but in this instance Elizabeth is NOT using "Lord" to mean her husband, or a member of the aristocracy, or a landowner. She is using it as it applies to Jesus in His true dignity, as St. Thomas said, "My Lord and My God."
This is not just a title, but a reality. It's not like a polite term of address what was fetched out of thin air. It comes from the Gospel, as you can see.
Elizabeth likely recognized Mary's baby as the promised Messiah. Jews did not believe Messiah would be God - still don't. For this reason, TC's commentary rightly says, "the Bible never ascribes the title "Mother of God" to Mary." It's earliest usage appears to be from Origen in about 250 AD.
By “Lord” Elizabeth meant Jesus, not the entire Godhead. Jesus is God, but not all of God is Jesus. Consequently the Bible never ascribes the title “Mother of God” to Mary. She was the mother of Jesus, who was Elizabeth’s Lord, since He was God. This usage does not necessarily imply that the person using it believed that Jesus was God. Elizabeth apparently meant that Jesus was the Messiah at least. Luke evidently used the term “Lord” frequently because for Greek readers “Christ” or “Messiah” had little meaning. The pagan Gentiles referred to Caesar as “Lord” Caesar meaning that he was their divine sovereign. “Lord” had the same connotation for Luke’s original readers. Jesus is the divine sovereign for Christians. - TC
... to be complete.
Your second example of Thomas’ statement “My Lord and My God.”
He is saying Master (Messiah) and God. Never ascribed to Mary. Jesus Christ, you bet!