Mary's last name is Tannenbaum. The joke is supposed to be that her name is Mary Christmas tree. Tannenbaum is a German word which is often mistakenly translated as Christmas tree, when it actually means Fir tree. The German word for Christmas tree is Weihnachtsbaum. The German song, O Tannenbaum is a Christmas song, and the English version is called, O Christmas tree, but O Tannenbaum is not a song about Christmas trees.
WILHELM SCREAM: Heard from the sound engineer in Bart's treehouse.
Mary's surgeon fiancée has no name. Throughout he is only ever referred to as "my surgeon fiancée."
Annie Potts gets mentioned in this episode and Richard Kind voices the movie director. Annie and Richard share screen time as love interests on Young Sheldon.