Is there something the authors intend to help the reader understand in several uses of “their” synagogue in comparison to “the” Temple? The word “their” is the possessive form of the third-person plural pronoun “they”; meaning “belonging to them”.
When referencing the temple, it’s often mentioned preceded by a definite article “the” which suggests uniqueness. To my understanding, the temple is not referred to as "their" temple.
Here are examples of “their" synagogue (KJV)
Matthew 12:9 And when he was departed thence, he went into their synagogue: nd, behold, there was a man which had his hand withered. And they asked him, saying, Is it lawful to heal on the sabbath days? that they might accuse him.
Matthew 13:54 And when he was come into his own country, he taught them in their synagogue, insomuch that they were astonished, and said, Whence hath this man this wisdom, and these mighty works?
Mark 1:23 And there was in their synagogue a man with an unclean spirit; and he cried out, (24) Saying, Let us alone; what have we to do with thee, thou Jesus of Nazareth? art thou come to destroy us? I know thee who thou art, the Holy One of God.
Uses of “the” temple.(KJV)
Matthew 21:12 And Jesus went into the temple of God, and cast out all them that sold and bought in the temple, and overthrew the tables of the moneychangers, and the seats of them that sold doves
Matthew 23:21 And whoso shall swear by the temple, sweareth by it, and by him that dwelleth therein.
Mark 12:35 And Jesus answered and said, while he taught in the temple, How say the scribes that Christ is the son of David?
In the gospel of Matthew originally written in Hebrew, the similar approach to the phrase is used but the words are different. The phrases used are: “their house of assembly” in comparison to “the Sanctuary”.