Yes, the Paladin's Fighting Style list is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are historically* chivalrous, honorable defenders of good / destroyers of evil that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

_{^{* in 5th edition, this isn't as strict anymore - previous editions had alignment restrictions on classes and the like, but 5e is a lot more open in many ways. Instead of always being the classic lawful good idealist, Paladins can also be evil; see this question, for instance. What stayed the same is that they are supposed to adhere to a strict code of some kind, and follow its principles.}}

That being said, while the explanation above reflects the RAW and RAI (rules as written/intended), you can always ask your DM if he can make an exception here, allowing you to pick a different fighting style for your Paladin. Mechanically, it shouldn't be an issue.

Other than that, there are also RAW-legal ways to gain access to non-Paladin-native fighting styles as a Paladin, which mostly (or all, I'm not sure) require multiclassing, as detailed in Someone_Evil's answer

No, Paladins cannot get the Two-Weapon-Fighting Fighting Style. The list in their class description is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are historically* chivalrous, honorable defenders of good / destroyers of evil that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

_{^{* in 5th edition, this isn't as strict anymore - previous editions had alignment restrictions on classes and the like, but 5e is a lot more open in many ways. Instead of always being the classic lawful good idealist, Paladins can also be evil; see this question, for instance. What stayed the same is that they are supposed to adhere to a strict code of some kind, and follow its principles.}}

That being said, while the explanation above reflects the RAW and RAI (rules as written/intended), you can always ask your DM if he can make an exception here, allowing you to pick a different fighting style for your Paladin. Mechanically, it shouldn't be an issue.

Other than that, there are also RAW-legal ways to gain access to non-Paladin-native fighting styles as a Paladin, which mostly (or all, I'm not sure) require multiclassing, as detailed in Someone_Evil's answer

Yes, the Paladin's Fighting Style list is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are chivalrous, honorable defenders of good that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

That being said, while the explanation above reflects the RAW and RAI (rules as written/intended), you can always ask your DM if he can make an exception here, allowing you to pick a different fighting style for your Paladin. Mechanically, it shouldn't be an issue.

Other than that, there are also RAW-legal ways to gain access to non-Paladin-native fighting styles as a Paladin, which mostly (or all, I'm not sure) require multiclassing, as detailed in Someone_Evil's answer

Yes, the Paladin's Fighting Style list is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are historically* chivalrous, honorable defenders of good / destroyers of evil that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

_{^{* in 5th edition, this isn't as strict anymore - previous editions had alignment restrictions on classes and the like, but 5e is a lot more open in many ways. Instead of always being the classic lawful good idealist, Paladins can also be evil; see this question, for instance. What stayed the same is that they are supposed to adhere to a strict code of some kind, and follow its principles.}}

That being said, while the explanation above reflects the RAW and RAI (rules as written/intended), you can always ask your DM if he can make an exception here, allowing you to pick a different fighting style for your Paladin. Mechanically, it shouldn't be an issue.

Other than that, there are also RAW-legal ways to gain access to non-Paladin-native fighting styles as a Paladin, which mostly (or all, I'm not sure) require multiclassing, as detailed in Someone_Evil's answer

Yes, the Paladin's Fighting Style list is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are chivalrous, honorable defenders of good that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

Yes, the Paladin's Fighting Style list is exhaustive.

This is apparent from the feature's description (emphasis mine):

At 2nd level, you adopt a style of fighting as your specialty. Choose one of the following options. You can't take a Fighting Style option more than once, even if you later get to choose again.

If you were supposed to choose from all fighting styles, then there would likely be a separate list in the PHB that isn't part of a class description. However, each class that has access to Fighting Styles lists its own selection of Fighting Styles - this is not a coincidence.

Fighters can choose from among all fighting styles, as fighting is literally in their name. They're designed to be viable for any kind of nonmagical combat - regardless of whether it's melee or ranged, and how you want to fight - well-armored, agile, two-handed, with a giant greataxe, etc.

Rangers, on the other hand, focus on dexterous combat, and especially ranged combat (duh). Their available Fighting Styles reflect this.

Barbarians are a dedicated melee class, but since their "theme" is rage-smashing stuff instead of elaborate fighting skills, they don't gain a Fighting Styles, but other features instead.

Paladins, lastly, are chivalrous, honorable defenders of good that are usually portrayed as melee combatants that gain access to a few spells. Their Fighting Styles reflect this similar to the Ranger, except their "theme" is not about dexterous two-weapon-fighting or archery - hence they don't get access to the respective Fighting Styles.

That being said, while the explanation above reflects the RAW and RAI (rules as written/intended), you can always ask your DM if he can make an exception here, allowing you to pick a different fighting style for your Paladin. Mechanically, it shouldn't be an issue.

Other than that, there are also RAW-legal ways to gain access to non-Paladin-native fighting styles as a Paladin, which mostly (or all, I'm not sure) require multiclassing, as detailed in Someone_Evil's answer