Matthew 17:24 New International Version
After Jesus and his disciples arrived in Capernaum, the collectors of the two-drachma temple tax came to Peter and asked, "Doesn't your teacher pay the temple tax?"
English Standard Version
When they came to Capernaum, the collectors of the two-drachma tax went up to Peter and said, “Does your teacher not pay the tax?”
In Matthew 17:24-27 are they paying to the Pharisees and the temple "lest we should offend them" and in Matthew 22:15-22 they are paying tax to Caesar "Render therefore unto Caesar the things which are Caesar's"
Matthew 17:24-27 King James Version (KJV) 24 And when they were come to Capernaum, they that received tribute money came to Peter, and said, Doth not your master pay tribute?
25 He saith, Yes. And when he was come into the house, Jesus prevented him, saying, What thinkest thou, Simon? of whom do the kings of the earth take custom or tribute? of their own children, or of strangers?
26 Peter saith unto him, Of strangers. Jesus saith unto him, Then are the children free.
27 Notwithstanding, lest we should offend them, go thou to the sea, and cast an hook, and take up the fish that first cometh up; and when thou hast opened his mouth, thou shalt find a piece of money: that take, and give unto them for me and thee.
Matthew 22:15-22 King James Version (KJV) 15 Then went the Pharisees, and took counsel how they might entangle him in his talk.
16 And they sent out unto him their disciples with the Herodians, saying, Master, we know that thou art true, and teachest the way of God in truth, neither carest thou for any man: for thou regardest not the person of men.
17 Tell us therefore, What thinkest thou? Is it lawful to give tribute unto Caesar, or not?
18 But Jesus perceived their wickedness, and said, Why tempt ye me, ye hypocrites?
19 Shew me the tribute money. And they brought unto him a penny.
20 And he saith unto them, Whose is this image and superscription?
21 They say unto him, Caesar's. Then saith he unto them, Render therefore unto Caesar the things which are Caesar's; and unto God the things that are God's.
22 When they had heard these words, they marvelled, and left him, and went their way.
Yes. While the temple was standing, this tax was used to fund the Jewish temple in Jerusalem.
Only after the temple was destroyed by the armies of Titus was this tax repurposed by the Romans to fund the temple of Jupiter Capitolinus in Rome. This was deeply aggravating to the Jews (see a summary of the history by Edmundson here).
As already noted by Daniel, the precedent for this 2nd temple era tax--before it was subverted by Rome--was described in the Torah (see Ex. 30:13)
Bonus material
This passage is one of the evidences that the Gospel of Matthew was written while the temple was still standing. It would have been absurd & insulting for the Jewish-Christian writer of this Gospel to include a favorable statement on the temple tax at a time when the tax was funding a pagan temple.
