Is the Prophet Jonah the same Jonah as in 2 Kings 14:23-25?

It is uncertain. They are probably different people.

According to archaeology, the Jonah who visited Nineveh came during the reign of Esarhaddon (681 BC). The Jonah from 2 Kings 14 was present in the reign of Jeroboam (901 BC). If this is correct, it could not be the same person.

The bible does not clearly state they are the same person. It is also possible the archaeology is mistaken.

Reference articles:

• Beneath Biblical Prophet's Tomb, an Archaeological Surprise

• Archaeologists find Biblical evidence under Jonah’s tomb

It is probable that they are the same person.

In 2 Kings 25, one reads

Jonah the son of Amittai the prophet, who was from Gath-Hepher

From the Faith Study Bible, in Jonah 1:1, there's the following note

Jonah was from Gath Hepher (meaning “winepress of the pit”), a town in lower Galilee near Nazareth (2 Kgs 14:25). The town has been identified with the modern village of el-Meshhed, where the tomb of Jonah is still shown. Gath Hepher was within the tribal territory given to Zebulun (see Josh 19:10, 13), so it is likely Jonah belonged to that tribe. Jonah ministered during the reign of Jeroboam II of Israel (793–753 bc; 2 Kgs 14:23–25).

And if they are the same person, as a curiosity, apart from the book of Jonah, the only OT reference to the prophet Jonah is precisely here in 2 Kings 14.

I am fairly certain they are the same because the reign of Jeroboam II are from 786BC to 746BC. That places Jonah during this same time.

Is the Prophet Jonah the same Jonah as in 2 Kings 14:23-25?

It is quite possible they are one in the same person. But historically it may be impossible to determine one way or another. Most sources are in favour of the interpretation that the two are one in the same person.

Book of Jonah, also spelled Jonas, the fifth of 12 Old Testament books that bear the names of the Minor Prophets, embraced in a single book, The Twelve, in the Jewish canon. Unlike other Old Testament prophetic books, Jonah is not a collection of the prophet’s oracles but primarily a narrative about the man.

Jonah is portrayed as a recalcitrant prophet who flees from God’s summons to prophesy against the wickedness of the city of Nineveh. According to the opening verse, Jonah is the son of Amittai. This lineage identifies him with the Jonah mentioned in II Kings 14:25 who prophesied during the reign of Jeroboam II, about 785 BC. It is possible that some of the traditional materials taken over by the book were associated with Jonah at an early date, but the book in its present form reflects a much later composition. It was written after the Babylonian Exile (6th century BC), probably in the 5th or 4th century and certainly no later than the 3rd, since Jonah is listed among the Minor Prophets in the apocryphal book of Ecclesiasticus, composed about 190. Like the Book of Ruth, which was written at about the same period, it opposes the narrow Jewish nationalism characteristic of the period following the reforms of Ezra and Nehemiah with their emphasis on Jewish exclusivity. Thus the prophet Jonah, like the Jews of the day, abhors even the idea of salvation for the Gentiles. God chastises him for his attitude, and the book affirms that God’s mercy extends even to the inhabitants of a hated foreign city. The incident of the great fish, recalling Leviathan, the monster of the deep used elsewhere in the Old Testament as the embodiment of evil, symbolizes the nation’s exile and return.

As the story is related in the Book of Jonah, the prophet Jonah is called by God to go to Nineveh (a great Assyrian city) and prophesy disaster because of the city’s excessive wickedness. Jonah, in the story, feels about Nineveh as does the author of the Book of Nahum—that the city must inevitably fall because of God’s judgment against it. Thus Jonah does not want to prophesy, because Nineveh might repent and thereby be saved. So he rushes down to Joppa and takes passage in a ship that will carry him in the opposite direction, thinking to escape God. A storm of unprecedented severity strikes the ship, and in spite of all that the master and crew can do, it shows signs of breaking up and foundering. Lots are cast, and Jonah confesses that it is his presence on board that is causing the storm. At his request, he is thrown overboard, and the storm subsides. - Book of Jonah

Wikipedia has the following to say about the Prophet Jonah:

In Judaism, the Book of Jonah (Yonah יונה) is one of the twelve minor prophets included in the Tanakh. According to one tradition, Jonah was the boy brought back to life by Elijah the prophet in 1 Kings 17. Another tradition holds that he was the son of the woman of Shunem brought back to life by Elisha in 2 Kings 4 and that he is called the "son of Amittai" (Truth) due to his mother's recognition of Elisha's identity as a prophet in 2 Kings 17:24. The Book of Jonah is read every year, in its original Hebrew and in its entirety, on Yom Kippur – the Day of Atonement, as the Haftarah at the afternoon mincha prayer.

In the Book of Tobit, Jonah is mentioned twice in the fourteenth chapter of the Book of Tobit, the conclusion of which finds Tobit's son, Tobias, at the extreme age of 127 years, rejoicing at the news of Nineveh's destruction by Nebuchadnezzar and Ahasuerus in apparent fulfillment of Jonah's prophecy against the Assyrian capital.

In the New Testament, Jonah is mentioned in Matthew 12:38–41 and 16:4 and in Luke 11:29–32. In the Gospel of Matthew, Jesus makes a reference to Jonah when he is asked for a sign by some of the scribes and the Pharisees. Jesus says that the sign will be the sign of Jonah: Jonah's restoration after three days inside the great fish prefigures His own resurrection.

The Prophet Jonah is regarded as a saint by a number of Christian denominations. His feast day in the Roman Catholic Church is on 21 September, according to the Martyrologium Romanum. On the Eastern Orthodox liturgical calendar, Jonah's feast day is on 22 September. In the Armenian Apostolic Church, moveable feasts are held in commemoration of Jonah as a single prophet and as one of the Twelve Minor Prophets. Jonah's mission to the Ninevites is commemorated by the Fast of Nineveh in Syriac and Oriental Orthodox Churches. Jonah is commemorated as a prophet in the Calendar of Saints of the Missouri Synod of the Lutheran Church on 22 September.

The Catholic Encyclopedia says the two are the same person.

Apart from the book traditionally ascribed to him, Jonah is mentioned only once in the Old Testament, 2 Kings 14:25, where it is stated that the restoration by Jeroboam II of the borders of Israel against the incursions of foreign invaders was a fulfillment of the “word of the Lord the God of Israel, which he spoke by his servant Jonah the son of Amathi, the prophet, who was of Geth, which is in Opher”. This last is but a paraphrastic rendering of the name Gath-Hepher, a town in the territory of Zabulon, which was probably the birthplace of the Prophet, and where his grave was still pointed out in the time of Saint Jerome. Mention is made of Jonah in Matthew 12:39 sqq., and in 16:4, and likewise in the parallel passages of Luke (11:29, 30, 32), but these references add nothing to the information contained in the Old Testament data. According to an ancient tradition mentioned by Saint Jerome, and which is found in Pseudo-Epiphanius, Jonah was the son of the widow of Sarephta whose resuscitation by the Prophet Elias is narrated in 1 Kings 17, but this legend seems to have no other foundation than the phonetic resemblance between the proper name Amathi, father of the Prophet, and the Hebrew Emeth, “truth”, applied to the word of God through Elias by the widow of Sarephta (1 Kings 17:24).